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PostSubject: Michel Serres Wed Jul 13, 2016 12:24 pm

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"ἐδιζησάμην ἐμεωυτόν." [Heraclitus]

"All that exists is just and unjust and equally justified in both." [Aeschylus, Prometheus]

"The history of everyday is constituted by our habits. ... How have you lived today?" [N.]

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PostSubject: Re: Michel Serres Thu Jul 14, 2016 11:58 am

Serres wrote:
"La Fontaine & Descartes    

The Wolf and the Lamb

The reason of the stronger is always the best.! We will show this shortly.
A Lamb quenched his thirst
In the current of a pure stream,

A fasting Wolf arrives, looking for adventure, And whom hunger draws to this place.
"Who makes you so bold as to muddy my drink?" Said the animal, full of rage :

"You will be punished for your temerity."
"Sire," answers the Lamb, "may it please Your Majesty Not to become angry;
But rather let Him consider
That I am quenching my thirst
In the stream,
More than twenty steps below Him;
And that, as a result, in no way
Can I muddy His drink."
"You muddy it," responded this cruel beast;
"And I know that you slandered me last year."
"How could I have done so, if I had not yet been born?" Responded the Lamb; "I am not yet weaned."
"If it is not you, then it is your brother."
"I do not have any." "Then it is one of your clan ;
For you hardly spare me,
You, your shepherds, and your dogs.
I have been told: I must avenge myself."


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Let there be for example three points A, B, and C on a line D, and a direction defined by the arrow. The ordering relation between these three points, which are elements of the set, can be one of "predecession" or of succes­ sion. A precedes B, which precedes C. C, in turn, is the successor of B, which succeeds A. One sees immediately that no point is its own prede­ cessor or successor: the relation is irreflexive. If, on the other hand, A precedes B, it is impossible for B to precede A ; the relation is antisym­ metric. Finally, if A precedes B and if B precedes C, then A precedes C: the relation is transitive. An ordering relation is irreflexive, antisym­ metric, and transitive. An ordered structure is a set provided with such a relation.

The "form of proces." This term has at least two meanings: the judicial meaning (trial), and the etymo­ logical meaning (process). A process includes a predecession and a suc­ cession: it is an order. Question: what is, first of all, the form of the trial, to wit, the form of the process? Here the form is a reason, a ratio, a con­ nection, a relation.

"The reason of the strongest" is definitely an ordering relation. A cannot be stronger than itself. A's being stronger than B excludes B's being stronger than A, and if A is stronger than B, and B is stronger than C, it follows that A is stronger than C.

Being stronger clearly defines an ordered structure. This is the first (we will call it the biological) model. The whole question will soon become one of finding the strongest, he who will have no predecessor in the order, but only successors.

Being "better" is also an ordering relation. A cannot be better than itself. A 's being better than B excludes B's being better than A ; if A is better than B and B is better than C, then A is better than C. We will call this second model of the ordered structure ethical. The whole question will soon become one of passing from the relative (an ordering relation) to the absolute, of finding the best, he who will have no predecessor in the order, only successors. The movement of the transitive relation is therefore blocked in order to arrive at stability, invariance: always. Finally, the use of is ("The reason of the strongest is always the best") indicates the invariance of the models in the structure, and therefore there is no need for demonstration : it is always a matter of the same process.

Let there be "the current of a pure stream." This is a third, topographi­ cal model of the same structure. It deals with an irreversible process which can, nevertheless, be determined at any point using an "upstream­ downstream" type of relation.
In the fourth place, in an irreversible stream, one can define a process of causality. The cause precedes the effect,'which succeeds the cause, without any possible reversal, without moving against the current. The third model was sequential ; this one is consequential.

Since the cause is upstream from the effect, the lamb replies: "And that, as a result, in no way / Can I muddy His drink." One finds here a demonstration. The demonstration by cause and effect is only one particular model of the global structural chain. The lamb demonstrates and La Fontaine shows. Whereas the latter shows the structural invariance using the model's variance, the former demon­ strates his point by using only one of the structure's models. Hence the idea, which can help us understand Descartes : the order of reason is only a particular exemplar of order in general. And this result has immense consequences.

One can construct a phenomenon on a spatial-type sequence or on a chain of consequences. Geometry, algebra, and physics constitute the Cartesian construct of the real. As Descartes wrote to R. P. Bourdin, the simplest of these phenomena can be seen in a basket of apples; if one of them is rotten, it diffuses rottenness around it by an irreversible process. In other words, and contrary to certain cosmogonies, the chaotic mixture succeeds separation, and impurity succeeds purity.

We have since learned that this belongs to the irreversibility principle of thermodynamics (the law of entropy). The chain of purity or separation followed by mixture is the physical model of the ordered structure. For us, it is isomorphic to the relation of the strongest: maximal energy is always upstream in an irre­ versible process. It is always a wolf. and not a lamb, who quenches his thirst in the traru;parent stream of a pure reason.

Now let us choose a political hierarchy, such as that of the classical age. Mark two points on our drawing and name them king and subject. This is a new model of the ordered structure : '' 'Sire,' answers the Lamb, 'may it please Your Majesty / Not to become angry; / But rather let Him con­ sider / That I am quenching my thirst / In the stream, / More than twenty steps below Him.'" Here there is something new. It is no longer the case of a strong individual who can find a still stronger one, of a "betterable better," an upstream that is downstream from another spot, a cause which can be an effect, or a purifiable energy; it is not, in short, the case of a greater, but of a maximum. There is nothing above the king. Is this the answer to our previous question?

The trial is a process whose global balance sheet can easily be recorded. It consists of an ordered structure with given axioms, a structure that branches out in several models: the social tree, the genealogical tree, the tree of time and history, the political tree, the tree of the production of energy, of entropy, and of pollution, the tree of causes, the hydrographic tree, the tree of the "better," the tree of good, evil, and knowledge, the tree of the distribution of forces - and a tree in general. All these trees together make a forest, into which "The Wolf carries him [the lamb] off, and then eats him."

This is not demonstrated by an order between that which precedes and that which follows, but shown as a forest of models, a forest of symbols. The proof is only one process among others: there exist philosophers from whom a whole forest is hidden by a single tree.

Now, he who is upstream, he who is greater, is responsible and loses. The minorant wins and eats the other. Whether dealing with drinking, eating, or dying, the succession of moves in the game follows the ordering relation : you are the stronger, I am the weaker; you are upstream, I am downstream; you are the cause, I am the effect; you muddy it, I cannot muddy it; and so on. The lamb shows, at every move, that he (or the third man) is absent from the upper position where his adversary places him. In short, the wolf "majorizes" or maximizes the lamb, who "minorizes" or mini­ mizes himself. Everything is played upstream from the wolf: however, are the places there occupied or vacant? And how is this going to de­ termine the results ofthegame?Theorem I: the lamb wins. The number of moves is almost infinite. There are as many of them as there are models of the ordered structure and, as a result, the game would never end: it would be necessary to show at every move that the place is vacant. This is what the lamb does. But, in addition, in the ultimate instance, he no longer proves the place's vacancy, but rather its inexistence, and the game is over. Not only is the place vacant, but there is no place. If the wolf is the king, "Sire," and "Majesty," he does not have a majorant. He is in an absolute position, like an absolute monarch.

Not only is there no third man, but it is impossible to conceive of one: quo nihil majus cogitari potest. Therefore the lamb has won, and the wolf has no majorant. He is himself the maximum. But then there is theorem II : the wolf carries him off, nonetheless, and he does it according to the rule of the game. He succeeds in showing the existence of a third man, upstream from himself, in the lamb's social group. This is because the shepherds and the dogs, protectors of the flock, are, in reality, much stronger than the wolf; they retain, upstream, the constant possibility of doing him harm. "I have been told": quo nihil majus dici potest. In the ordering relation, they are clearly majorants. The place preceding the wolf's place is occupied by the shepherd, who is the strongest. The shepherd and his watchdogs are above the "king-wolf." The fable is a perfect operational definition - per­ fect in that it is free of all psychologism-of hypocrisy. In fact, the term hypocrisy comes from the verb to judge, to choose, to decide, and from the prefix underneath. In other words, if you want to win, play the role of the minorant.

Stable structures and dialectical processes are inseparable.

Besides, let us note the circle: A is upstream from B, A must place B or a third person upstream from himself in order to have the right to eat or kill the adversary. Let us, for the moment, retain the three results : ordered structure, fight to the death, and circularity.

One commands nature only by obeying it. This is probably a political ideology-betrayed by the prosopopeia - which implies practices of ruse and subtlety : in short, a whole strategy. Since nature is stronger than we are, we must bend to its law, and it is through this subterfuge that we dominate it. We are under its orders and turn its forces back against order. This is the circle of ruse and productive hypocrisy: nature is a majorant; we try, ourselves downstream, to majorizet ourselves in relation to it. Here one finds again, intact, an ordered struc" ture, a game, its rules (and how best to implement them), the struggle to seize power, and the closed cycle outlined by these moves.

Descartes, after Bacon, picks up the precept: he calls for us to become the masters and possessors of nature. The impulse to obey has just disap­ peared. Baconian physics made science into a duel, a combat, a struggle for domination ; it gave it an agonistic model, proposing a form of ruse for it so that the weak party would triumph. It transformed science into a game of strategy, with its rules and its moves. But Baconian reason is a weak reason which loses at least the first round, because it first resigns itself to obedience. Descartes rejects this, and, consequently, he suppresses the loss. In the relationship of agonistic forces between ourselves and the exterior world, he seeks the means that will permit us to win at every move. "The reason of the strongest is always the best." The best reason always permits a winning game.
The foundation of modern science is in this word, always. Science is a game, an infinite game, in which we always win. Reason is an absolute and constant "optimization."

In a contest, a competitor is not always assured of winning. A player stronger at a given moment because of a given move can later fail when his opponent discovers the means or obtains the power to pass upstream from him. The dichotomy then appears to reverse itself; the weaker has taken the stronger's place. In fact, it is the entire couplet which is displaced in the game-space structured by the ordering relation. This displace­ ment is infinite and does not stop-as long as one stays in the same space - since it is relative. It is the infernal time of hierarchical struggle, the time of human unhappiness. There are two, and only two, strategies that can give a final turn to the sequence of moves. First, one stays with the dialecticalgame and tries to discover a martingale in order to win, whatever the move might be: then the game is over and there is a de.: fin,itive dominant. Old times are over and struggles stop under the in­ surmountable power of one of the contestants. With a maximal move, one freezes the game-space in a single pattern of order and hierarchy. It is the end of a slice of history. Second, one attacks the ordered structure itself- which is the condition for the game's existence or, rather, without which the game can have neither space nor time-in order to shatter it. This move would mark the beginning of a new history. Philosophers have rarely taken the second path: they have always tried to find the maximum and the minimum points at the edge of the space organized by the couplet of the majorant and the minorant. As soon as it is discovered, one can say : always. And it is always the time of the wolf.

Look at Rousseau, for example. He repeats, after many others: the stronger is never strong enough always to be the master unless he trans­ forms his might into right and obedience into duty. As we indicated earlier, this kind of transformation is the shift from one model to the other: another move, same game. The second move is as unstable as the first: jurisprudence and ethics are relative to a cultural space organized by the ordering relation. At times a radical, at others a tiny, change in the ordering relation is sufficient to make an entire group overthrow its morals and its laws. The trial's dialectics remain, based on the majorant's and the minorant's relationships, with the division of the stakes left to the balanced distribution of forces and to the recuperation of ruse. It is therefore necessary to recognize an infinity of moves in the relative field of the "more" and the "less." As in the fable, one must maximize the "more" and minimize the "less." One must maximize in an absolute fashion, in such a way that there may not exist, that one may not conceive, a majorant to a maximum and a minorant to a minimum. One must trans­ form force into factual necessity and obedience into an inevitable law. One may cut off the king's head, kill the dog, or eat the shepherd, yet one cannot do without Reason's verdicts. And this is why, since Rousseau, one no longer hesitates to invoke science in the realm oflaw, power, and politics. It is because science has already pointed the way to the winning strategy. For it must be remembered that the foundation of science-whether it be the pure sciences at the Hellenic dawn or the experimental sciences in the classical age - had taken place in an agonistic field.

One can show that abstract mathematics and axiomatics owe their emergence to the Sophists' discussions and paradoxes, as well as to Plato's dialogue techniques. Agonistics is there, in the background. And yet the purest positivist cannot challenge Auguste Comte's analysis, which defines the birth of geometry (in his eyes a natural science) as a ruse or set of ruses: to be able to measure inaccessible things, to find indirect means for man to perform that which he does not have the means to do. Once again, this is a strategy. And as soon as laws are written, they allow man always to have access to the inaccessible. The stability and constancy of certitudes or precisions are conceived in the beginning as the end of a prior game.

Another founding word was that of Galileo: nature is written, it is drafted in a language; everyone agrees that this is a mathematical lan­ guage. But this writing is not obvious, it is hidden, concealed under the phenomenal appearance of the material world. One must force open the secret, find the key to the logogriph, and decode this writing. Now, in this game of decoding or deciphering, nature defends itself. It is subtle, it is hidden, it is secret. One must therefore employ subtler strategies in order to make its defenses fail. Once the key is discovered, the world surrenders.

Just as in Plato's work there abound traces of this state of affairs neces­ sary for the founding of the rigorous sciences, so, in the same way, Des­ cartes's work shows such traces at the dawn of exact sciences (conceived, since the classical age, as the optimal relationship from subject to object). I have recalled this founding word at the end of which we should have made ourselves the masters and possessors of nature. And I expressed it in terms of a game: Baconian obedience having been suppressed, the project became one of always winning. Reason is optimized, it is the best, it is always invincible. From La Fontaine spring Descartes and the game, or vice versa-it matters little. The three elements located in the fable should then be found in the Metaphysical Meditations: a space structured by the ordering relation, a circle, a game with its moves, its end, and its winner. Two and only two have been recognized by the commenta­tors; the third, which is the most visible-since it concerns action-re­ mains hidden.

In the fable, one saw, quite simply, that if the direction of the moves remained at the level of the formal pair majorant-minorant, the game was endless and without a stable victor. It is therefore necessary to put an end to this once and for all; one of the adversaries must be assured of always winning. That is possible only if one passes from the position of majorant to a maximum without conceivable predecessor, and from the position of minorant to a minimum without any imaginable successor. There is no place above the king, there is no place above the shepherd assisted by his dogs, and there is no place below the lamb. From this comes the global theorem: in the Cartesian Meditations, all the moves are maximized.

Descartes again, speaking of doubt: "as much as reason persuades me already that I should "no less carefully" keep myself from believing in things that are "not entirely" certain and indubitable, "any more than" in those that appear to us to be "manifestly" false" (p. 405). Result: the universal quantificator. A constant repetition of all, always, never, absolutely, and so on . Appearances of always, the key word, "I shall always follow this path" (p. 414).

Quantification, until now, has been rather indefinite. Observe the pro­ gression from the first Meditation to the second: ""Any" subject for doubt that I find will suffice to make me reject "all" [opinions]" (p. 405); "it is "never entirely" prudent to trust those who have deceived us "once"" (ibid.); and "distancing myself from "everything" in which I will be able to imagine the "least" doubt" (p. 414). First we move from the universal (all) to the par­ ticular (any), then, to the reduction of the particular to a single case, (once), and finally, to the reduction of unicity to the minimum (the least). This is clearly the final move.

God's position and that of the atheists establish the rule: ""the less" powerful the author that they assigned to my origin will be, the more probable it is that I am so imperfect that I am "always" in error" (p. 410). It will suffice to envisage the extreme case in order to invert the result, to find the quo nihil cogitaripossit, sovereignly omnipotent, veracious. As far as I know, "perfect" signifies "optimal."

The global description of the procedure follows: "having so balanced my [new and old] prejudices that they can "no longer" sway my opinion" (p. 411). With the model of a simple machine, taken up again, later, at Archimedes' point (p. 414) (thus the minimum, to move the earth, the maximum), one obtains the static comparison of relationships. In this space, the optimized move is precisely the Archimedian fixed point. The progression is the same.

Speaking of the evil spirit, Cartesian progression is still the same: first called ""no less" wily and deceiving than powerful" (p. 412), the evil spirit is called later in the second Meditation "a very powerful and very wily de­ ceiver, who employs "all" his energy to deceive me "always"" (p. 415). We move again from the comparison of relationships to the maximal relationship such that nothing can exist beyond it. Here is the strategy in relation to this spirit: "I shall prepare my mind so well against "all" of this great de­ ceiver's ruses, that, "no matter how" powerful and wily he may be, he will never be able to impose "anything" on me" (p. 412). And the final move as Descartes sees it: "let him deceive me as much as he wishes, he will never manage to turn me into "nothing", as long as I think that I am "something"" (p. 415). This doubt is called hyperbolic, but no effort is made to under­ stand the hyperbole's function.Hypocritical ruse and hyperbolic doubt are operators totally devoid of psychologism.

"My meditation of yesterday has filled my mind with "so many" doubts, that it is "no longer" in my power to forget them . . ." (p. 414); "I am so surprised, that I cannot fix my feet on the bottom nor swim . . ." (ibid.).

The existence of the "I," "I am," "I exist" is clearly uncovered by a mini­ mum-maximum move: it is the minimal remainder of a maximized strategy or ruse. At the end of which, as soon as "everything" that can be in "any way" disputed has been dismissed, I [Descartes] obtain "a "more' certain and "more" evident knowledge than "all" the knowledge I had earlier (p. 416). Once again, the universal quantificator is the final move in the quantification of a relationship followed to its limit.

One could continue the demonstration. The syntax is constructed entirely in this way. The process is everywhere quantified, tactics are everywhere maximized, the final move is on the maximum maximorum, and even more on the quo nihil. . . . Not only is there no one in the places upstream, but there is no longer any upstream locus. To give oneself an adversary and defeat him with the help of an all-powerful and truthful associate, God Himself: this is a game between two players, between three, in which nature disappears-burned, melted, minimized, de­ stroyed. The malleable wax and I become one; thus I always win. God is a point without an upstream, the wax a point without a downstream, and myself in the center, hence the circle ; I can no longer lose at this game.

Then everything becomes possible : optics and dioptics, the world and its system, medicine and everything that follows from it. In the game of truth, error has been checkmated; in the game of domination, all is reduced to slavery, including the body. Metaphysics is operatory, it is the strategic set without which physics and the exact sciences are nothing but partial and dispersed tactics. Einstein rediscovered Descartes by turning around a parable : God is subtle, but he does not cheat. To know nature is a game. Not a futile amusement, but a deadly dangerous game. Nature's secret lies in the fact that one sees only the backs of the cards, and that one must play carefully and cautiously, in order to uncover this secret and read the faces of the cards, that is to say, to read them mathe­ matically. Experimentation is a game in which the more one cheats, the less one knows (hence morals and deontology), a game one can lose and win, but in which there exists a guaranteed winning strategy. The de­ velopment of mathematics, independent of experimentation, is another result: one must try to refine strategies, which are useful against an ad­ versary whose strategies are also extremely refined. "Game," then, is not just a word of science, it is the model of all exact knowledge. Informa­ tion theory, the daughter of physics and probabilities, has discovered this model once again. But during the classical age, it is a martial game. Like many other philosophers, Descartes pursued his military calling in metaphysics.

It is often said that probability theory and the art of conjecture were born, in a given economic context, from the idea pf life annuities, before the large banks and companies thought of insuring against death. This is probable, although not proven by the facts. Leibniz, among others, computed life annuities.

The more significant idea is that of the wager, a wager that is not very specific, since every martial game is a game to the death, a wager on death. If it is a question of dates, you have insurance and annuities; if it is a question of stakes, you have Pascal. Thus it is that the relation between theory and practice, the relation of metaphysics to knowledge, and the relation of the latter to domination come together in the same place, at the outcome provided by death.

For Plato and a tradition which lasted throughout the classical age, knowledge is a hunt. To know is to put to death - to kill the lamb, deep in the woods, in order to eat it. Moving from combat with prey outside the species to killing inside the species, knowledge now becomes military, a martial art. It is then more than a game; it is, literally, a strategy. These epistemologies are not innocent: at the critical tribunal they are calling for executions. They are policies promulgated by military strategists. To know is to kill, to rely on death, as in the case of the master and the slave.

Today we live out the major results of these wolfish actions. For the "I," who played the role of the lamb by minimizing his powers and placing the declared powers upstream from himself, this "I" is the wolf. In the ordering relation, in the game-space, the "I" is clearly in the middle, between the victorious sheepdog and the defeated devil or the wax. It has taken the wolf's place, its true place. The reason of the strongest is reason by itself. Western man is a wolf of science." [Hermes]

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"ἐδιζησάμην ἐμεωυτόν." [Heraclitus]

"All that exists is just and unjust and equally justified in both." [Aeschylus, Prometheus]

"The history of everyday is constituted by our habits. ... How have you lived today?" [N.]

*Become clean, my friends.*
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PostSubject: Re: Michel Serres Thu Jul 14, 2016 12:16 pm

Modernity vs. the Oedipal Incest of knowl-Edge


Serres wrote:
"The bridge is a path that connects two banks, or that makes a dis­ continuity continuous, or that crosses a fracture, or that patches a crack. The space of an itinerary is interrupted by a river; it is not a space of transport. Consequently, there is no longer one space; there are two without common boundaries. They are so different that they require a difficult, or dangerous, operator to connect their boundaries-difficult since at the very least a pontiff is necessary, dangerous since most of the time a devil of some sort stands watch or the enemies of Horatius Cocles stand ready to attack. Communication was interrupted; the bridge re-es­ tabiishes it vertiginously. The well is a hole in space, a local tear in a spatial variety. It can disconnect a trajectory that passes through, and the traveler falls in, the fall of the vector, but it can also connect spatial varieties that might be piled upon one another : leaves, layers, geological formations. The bridge is paradoxical : it connects the disconnected. The well is more paradoxical still: it disconnects the connected, but it also connects the disconnected. The astronomer falls in (Thales); the truth comes out. The killer dragon lives there, but one draws the water of immortality from there.

And suddenly, I am speaking with several voices; I can no longer draw the line between narrative, myth, and science. Is this bridge the Konigsberg bridge where Euler invented topology, a bridge over the Viorne or the Seine in the Rougon-Macquart cycle, or the whole group of bridges revealed in mythical discourses? No, I no longer have the choice, and it is the same bridge. Is this well a hole in Riemannian spatial varieties, a well of potentiality in which, at its lowest ebb, appears the germinating point, as in Thorn, or the Plassans well, or Jacob's? No, I no longer have the choice, and it is the same well. In every case, and so much the worse for classification, connec­ tion and non-connection are at stake, space is at stake, an itinerary is at stake. And thus the essential thing is no longer this particular figure, this particular symbol, or this particular artifact; the formal invariant is some­ thing like a transport, a wandering, a journey across separated spatial varieties. Circumnavigation of Ulysses or of Gilgamesh and topology.

My body (I cannot help it) is not plunged into a single, specified space. It works in Euclidean space, but it only works there. It sees in a projective space; it touches, caresses, and feels in a topological space; it suffers in another; hears and communicates in a third; and so forth, as far as one wishes to go. Euclidean space was chosen in our work-oriented cultures because it is the space of work-of the mason, the surveyor, or the architect. Hence the cultural idea of the practical origins of geometry that is a tautology, since the only recognized space is preciselv that of work,_of tranport. My body, there­ fore, is not plunged into a single space, but into the difficult intersection of this numerous family, into the set of connections and junctions to be established between these varieties. This is not simply given or is not always already there, as the saying goes. This intersection, these junctions, always need to be constructed. And in general whoever is unsuccessful in this undertaking is considered sick. His body explodes from the discon­ nection of spaces. My body lives in as many spaces as the society, the group, or the collectivity have formed: the Euclidean house, the street and its network, the open and closed garden, the church or the enclosed spaces of the sacred, the school and its spatial varieties containing fixed points, and the complex ensemble of flow-charts, those of language, of the factory, of the family, of the political party, and so forth. Conse­ quently, my body is not plunged into one space but into the intersection or the junctions of this multiplicity. Again, whoever fails or refuses to pass like everyone else through the crossroads of these multiple con­ nections-whoever remains in one of these spaces, or, on the contrary, refuses all of them -is treated as ill-adapted or delinquent or disoriented. Such is the case, for example, for whoever remains frozen, hung up, in the family tree, whoever fears leaving a closed paradise between two branches of a river, or whoever wants to tear apart the network which he endures as he would a prison or slavery's iron shackles. This brings us to the beginning. The fact is that in general a culture constructs in and by its history an original intersection between such spatial varieties, a node of very precise and particular connections. This construction, I believe, is that culture's very history. Cultures are differentiated by the form of the set of junctions, its appearance, its place, as well as by its changes of state, its fluctuations. But what they have in common and what constitutes them as such is the operation itself of joining, of connecting. The image of the weaver arises at this point: to link, to tie, to open bridges, path­ ways, wells, or relays among radically different spaces; to say (dire) what takes place between them; to inter-dict (inter-dire). The category of be­ tween is fundamental in topology and for our purposes here: to inter­ dict in the rupture and cracks between varieties completely enclosed upon themselves. "Enclosed" means isolated, closed, separated; it also means untainted, pure, and chaste. Now, that which is not chaste, incestus, can be incest. The incest prohibition (inter-diction) is, then, literally a local singularity exemplary of this operation in general, of the global project of connecting the disconnected, or the opposite, of opening what is closed, or again the opposite, and so forth.

Oedipus wanders and journeys, having set out from the palace of King Polybus to seek counsel from the Delphic oracle. At a crossroads he en­ counters Laius, his father, and Polyphontes, his father's herald, whom he kills. The crossroads is precisely the sought-for singularity. The roads for Daulis and Thebes meet at Megas; they form the road that rises toward Delphi through the valley. At Megas, the bifurcation. This is a very good point of departure, since in a diagram the example is trivial. There a road passes between two high rocks, as in a crevice or a narrow defile. Crossroads: cross, passage of a road across a ribbon that divides space, passing over a crack. Bridge : connection through the disconnected. To the left, ignorance, blindness, or the unconscious-the unknown and the unsaid. To the right, knowledge, the conscious, the sacred, Delphi and the signifier, the word. Oedipus is driven back from the narrow defile by Laius' team of horses, insulted by Polyphontes. The fact that the murder of the father takes place at this cross, this interrupted, joined edge, this limit or fault, is a catastrophe. Thus the circumstance is the murder and the law is traced upon the ground. To cross the broken threshold of the word. The essential thing is indeed the bifurcation. As soon as the father is involved, once again we come back to the law, a bifurcation traced on the family tree: father, mother, son, here again on the graph is inscribed the triviality of the narrative. To the left the one, to the right the other, and incest, we have already seen, is still another connection upon the disconnected. The text turns inside out like a glove and shows its function: the establishment of separations between spaces and their difficult junction. One can say that Oedipus kills Laius at this place, and miss the place, and thus repress the place of the repressed; or one can say instead that this place is such that Oedipus kills his father there, that it is a point so catastrophic and so confined that he must kill father and mother to go past it. To be the son or to place oneself at the crossroads: two bifurcations and two catastrophes that the myth joins together by its very word. Furthermore, the fact that the son's name is Oedipus repeats the same law. How can one move about in space when one's feet are afflicted? Now, to prevent him from journeying, his parents hang him feet up in the air. Oedipus regains his feet, he sets out for Delphi, the myth regains its feet.

The Sphinx, then, and the same law is repeated. This watchdog of Thebes dies as the result of a solution and lives as a result of solutions of continuity. She keeps close watch on the closed road where Thebans no longer pass, where they are devoured, in pieces. She is a chimera, half­ lion and half-woman; half four-legged, also, and half two-legged, and perhaps partly bird. She is a body sewn back together, badly sewn: two parts related by dichotomy, joined in the form of a Chl crowned by wings; she is a crossroads, with wings that protrude for one who no longer needs feet. The Sphinx is a bifurcation, and conversely. And the crossroads is a chimera. Thus everything is repeated, enigma and knowl­ edge, on the road to Thebes and the road to Delphi, catastrophe and passage, tear and connection. Oedipus is indeed the last descendant of the Sparto}, of disseminated spaces, of catastrophic separation, of the continuous that must be recovered. Everything is repeated once again when Jocasta recognizes her son by the scar on his feet, a scar in which the lips of a crevice connect. Now, Sophocles gives another version of the recognition scene, and his translation is faithful-Oedipus recognizes himself as a murderer at the moment in the narrative when Jocasta, the mother, mentions the crossroads, the chi. It is not I but Sophocles and the son and the mother together who draw the law out of the discourse.

From the beginning of the world protrayed in Plato's Timaeus, after reference to the chara, matrix and mother, in which we recognize a topological space, the Same and the Other, separated, are rejoined by the Demiurge in the figure of a chi. This figure is formed by the inclination of the ecliptic on the equator; the world is a chimera. The space of the world is described as requiring artful connection.

Now, then, at a certain beginning of a certain story, on the family tree containing ordered paths structured by some ordered relation, incest describes a loop that turns back upon itself toward a previous crossroads and strongly reconnects the spatial complex.

From this results the general and simple idea that mythical spaces are chimerical. This is a theorem containing a literal tautology but which uncovers a complicated state. Parts as separate as the Same and the Other are to be joined. Oedipus' itinerary crosses spatial accidents, bifurcations, catastrophes, and loops. Oedipus' discourse (discours) is identical to this itinerary (parcours). It poses chi's on cracks, crossroads between spatial varieties that do not have common boundaries. This in turn presupposes that before it, in other words, before discourse, there existed a multi­ plicity of unrelated spaces : chaos.

It would be necessary to demonstrate the generality of the hypothesis. The theme of the Odysseus cycle is not space, this discrete unit redis­ covered indefinitely or by repetitions along its discursive sequence. The plurality of disjointed spaces, all different, is the primal chaos, the condi­ tion of the series that assembles them. Ulysses' journey, like that of Oedipus, is an itinerary. And it is a discourse, the prefix of which I can now understand. It is not at all the discourse (discours) of an itinerary (parcours), but, radically, the itinerary (parcours) of a discourse (discours), the course, cursus, route, path that passes through the original disjunction, the bridge laid down across crevices. And the separation is of an im­ pregnable rigor.(All the spaces encountered are perfectly defined, without waver or blur. It'ls impossible to connect them among themselves. They cannot be composed to form a single homogeneous space. They combine such categories as open and closed, exterior and interior, boundary and limit, vicinity and adherence, and so forth, all concepts characteristic of the numerous spaces of topology. Hence comes everything one might desire in the text: inaccessible islands, and countries from which one cannot escape; the beach upon which the catastrophe casts you; the breaking of the waves; the shores from which one is hurled as one ap­ proaches. The intrusion of a wooden horse into the heart of the enclosed eitadel, where the warriors are at the same time inside the city but outside it by being inside the closed compartment that is inside the closed citadel. The exit of a ram, this bridge, out of an enclosed cavern in which a dan­ gerous fire burns; ram, horse anew, with the difference that the space of touch full of voids is more important here than optical or visual space.9 Hence the blindness of the Cyclops, in order to demonstrate that a closed system is not the same for the clairvoyant and for one who is reduced to his sense of touch.

Likewise, the attractive passage by the Sirens' shore where a vicinity, an adherence, is skirted, open for the deaf and closed for every listener. Original spaces proliferate on the map of the journey, perfectly disseminated, or literally sporadic, each one rigorously de­ termined. The global wandering, the mythical adventure, is, in the end, only the general joining of these spaces, as if the object or target of dis­ course were only to connect, or as if the junction, the relation, constituted the route by which the first discourse passes. Mythos, first logos; transport, first relation; junction, condition of transport. Thus we have Penelope at the theoretical position: the queen who weaves and unweaves, the originally feminine figure who, become male, will be Plato's Royal Weao;er. As Descartes says in Rule X, a tapestry intermingles threads with infinitely varied nuances.!o Infinitely: the rational and the irra­ tional. Descartes says this of a barbarous mathematics. Here we are once again. Barbarous or feminine, the logos is present, but still at the level of the hands. They connect. Penelope is the author, the signatory of the dis­ course; she traces its graph, she draws its itinerary. She makes and undoes this cloth that mimes the progress and delays of the navigator, of Ulysses on board his ship, the shuttle that weaves and interweaves fibers separated by the void , spatial varieties bordered by crevice s . She is the em­broideress, the lace-maker, by wells and bridges, of this continuous flux interrupted by catastrophes that is called discourse. In the palace of Ithaca, Ulysses, finally in the arms of the queen, finds the finished theory of his own mythos. The heroine of La Debacle, on the contrary, finds along her catastrophic route the weaver whose loom has just burned.

No one leaves here and no one enters-except those who speak geometry, the discourse that has com­ munication as its goal. Myth attempts to transform a chaos of separate spatial varieties into a space of communication, to re-link ecological clefts or to link them for the first time: from the mute animal to the proto-speaker. At the theoretical position in universo is she who conditions and who prepares the work of the weaver herself, she who produces and who gives the thread: Ariadne.

This can be general. All of Greece about which I am speaking is Dichotomy, Polytomy: Zeno's paradox, the Platonic classificatory trees, the division of segments by relations and proportions in the Euclidean manner, logos and analogy, the sharing of riches on Aristotle's scale, to each his part, his destined part. . . . This unitary discourse through dis­ tinctions and partitions, this discourse of the beginnings of mathematics, miraculously established, flows back toward its Pythagorean origin where the speakable, namely, the rational, is the split whole that we call fractions, the set of numbers that are the very things themselves. Here the method, road, path, track propose and set forth medians: the middle term between two terms. The completion of an interval is a problem that has not varied from the dawn of time up to Cantor. Let this bridge be lost and the endless nature of the path or the inaccessibility of the opposite shore be discovered, and we have the crisis, the shipwreck of Hippasus of Meta­ pontum, he who can no longer cross the sea. No one can speak any longer, and we have the irrational or the unspeakable-the incommu­ nicable, to be very precise. In fact, we have the return to the state of things before the establishment of rational discourse, the time when spaces were poorly joined, when transport and itinerary were only myth. The sect is dissolved when faced with the infinitely divisible-until the atomism of Democrites. Hippasus is shipwrecked like Ulysses, both of Metapontum (meta-pons, "metabridge"). Pythagoreanism had turned its back upon barbarous topology; it founders once again in myth with the discovery of the topology of real numbers. It had established a space of mediations, of communication, and it dies from losing it.

All this was rational, discursive, and speakable, all this was mathematical and logical, but in the closest vicinity to the sources, to the possibility of speaking to one another. The Greek cities were dispersed, reciprocally closed insu­larities, islands separated like the Sporades, in which every man worthy of the name, in other words, measure of all things, was inside, while on the exterior of this political space animals, barbarians with growling languages, circulated in a chaotic multiplicity of sociopolitical spaces: the world before its formation, the practical world before the emergence of scientific knowledge. This logos was first myth, in order to succeed in creating at least one koine.12 All the principles of the Greek cities go beyond this arm of the sea, before Troy, in order to found a language of communication-that the gods first make possible. The gods are en­ countered as the same-here, everywhere-because in their other space they enjoy a single space. It is essentilotl that one no longer know where Dionysus was born, where Oedipus and Theseus died. Anywhere : this is far preferable. Thus, in this discourse, chaos begins again: scattered members, the diasparagmos, the bones of Mother Earth, the first family of Spartol, dissemination in space, or, rather, dissemination of mor­ phologies themselves. Whereupon the first problem: to find the single space or the set of operators by which these spatial varieties in impracti­ cal, inconceivable vicinity will be joined together. To open the route, way, track, path in this incoherent chaos, this tattered cloud, whose dichotomic thicket is reformulated in the common space of transport when it is reconstructed. To find the relation, the logos of analogy, the chain of mediations, the common measure, the asses' bridge; to find the equilibrium or the clinamen.
The clinamen is the minimum angle to the laminar flow that initiates a turbulence".
Second answers, second words, where measure and correct measure presuppose a homogeneous space which is posited as reference and which is the answer to the first question asked: the unitary space of possible transports or of always possible transfers.

And thus one must find first, find conditionally, a word, a logos, that has already worked to connect the crevices which run across the spatial chaos of disconnected varieties. One must find the Weaver, the proto-worker of space, the prosopopeia of topology and nodes, the Weaver who works locally to join two worlds that are separated, according to the autochton's myth, by a sudden stoppage, the metastrophic caesura amassing deaths and shipwrecks: the catastrophe. He works, according to Plato, in a discourse where rational dichotomy and the myth of the two space-times, common measure and the Weaver, all converge. He untangles, interlaces, twists, assembles, passes above and below, rejoins the rational, the irra­ tional, namely, the speakable and the unspeakable, communication and the incommunicable. He is a worker of the single space, the space of measure and transport, the Euclidean space of every possible displace­ ment without change of state, royally substituted one fine day in place of the proliferating multiplicities of unlinked morphologies.

In order to practice dichotomy and its connected paths, one must know that its clefts follow and overlap the ancient mythical narrative in which worlds are torn asunder by a catastrophe-and only the Weaver knows how to link them again or can reunite them. Then and only then geometry is born and myth falls silent. Then the logos or relation unfolds, the chains and networks on the smooth space of transport, which itself alone replaces the discourse (discours) of itineraries (parcours). Linked homogeneity erases catastrophes, and congruent identity forgets difficult homeomorphisms. Reason, as the saying goes, has triumphed over myth. No, it is Euclidean space that has repressed a barbarous topology, it is transport and displace­ ment without obstacles that have suddenly taken the place of the journey, the ancient journey from islands to catastrophes, from passage to fault, from bridge to well, from relay to labyrinth. Myth is effaced in its original function, and the new space is universal, as is reason or the ratio that it sustains, only because within it there are no more encounters. As Plato says, one can walk there on two or four legs, follow the diagonal, freely choose the longest or shortest road, route, ode, or period, and so on, as much as one wishes. The earth is measured (geo-metry) by means of just measure (the King). The multiplicity, the dangerous flock of chaotic morphologies, is subdued. Thus the Statesman is written.

Hence the two great vicissitudes of the nineteenth century. Beneath the apparent unity of Euclidean space, mathematics, turning back toward its origins, rediscovers the teeming multiplicity of diverse and original spaces-and topology emerges as a science. We have not finished nor shall we ever again finish dealing with spaces. At the same moment, in an aged Europe asleep beneath the mantle of reason and measure, mythology reappears as an authentic discourse. The coupling of these rediscoveries becomes clear : Euler's bridge and the vessels' bridge across the Hellespont during the storm, Listing's or Maxwell's complex and the Cretan maze. Let us not forget that Leibniz, proto-inventor of the new science, said in time and against his time that one should listen to old wives' tales." [Hermes]

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"ἐδιζησάμην ἐμεωυτόν." [Heraclitus]

"All that exists is just and unjust and equally justified in both." [Aeschylus, Prometheus]

"The history of everyday is constituted by our habits. ... How have you lived today?" [N.]

*Become clean, my friends.*


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PostSubject: Re: Michel Serres Thu Jul 14, 2016 12:25 pm

The Law of Contradiction.


Serres wrote:
"The ca­cophonous speaker and the auditor, exchange their reciprocal roles in dialogue, where the source becomes reception, and the reception source (according to a given rhythm). They exchange roles sufficiently often for us to view them as struggling together against a common enemy. To hold a dialogue is to suppose a third man and to seek to exclude him. a successful communication is the exclusion of the third man. The most profound dialectical problem is not the problem of the Other, who is only a variety -or a variation-of the Same, it is the problem of the third man. We might call this third man the demon, the prosopopeia of noise.

For example, the Metaphysical Meditations can be ex­ plained according to these principles: the Meditations seek out the other with whom one must join in order to expel the third man. Dialectic makes the two interlocutors play on the same side; they do battle together to produce a truth on which they can agree, that is, to produce a successful communication. In a certain sense, they struggle together against interference, against the demon, against the third man. Obviously, this battle is not always successful. In the aporetic dialogues, victory rests with the powers of noise; in the other dialogues, the battle is fierce - attesting to the power of the third man. Serenity returns little by little when the exorcism is definitively(?) obtained.

The subject of abstract mathematics is the "we" of an ideal republic which is the city of communication maximally purged of noiselO (which, parenthetically, shows why Plato and Leibniz were not idealists). In general, to formalize is to carry out a process by which one passes from concrete modes of thinking to one or several abstract forms. It means to eliminate noise as well, in an optimal manner. It means to become aware of the fact that mathematics is the kingdom that admits only the absolutely unavoidable noise, the kingdom of quasi-perfect communication, the manthdnein, the kingdom of the excluded third man, in which the demon is almost definitively exorcised. If there were no mathematics, it would be necessary to renew the exorcism.

With Plato, on the contrary, the discussion is full and complete : it makes the recognitoin of the abstract form and the problem of the success of the dialogue coexist. When I say "bed," I am not speaking of such and such a be?, mine, yours, this one or that one; I am evoking the idea of the bed. When I draw a square and a diagonal in the sand, I do not in any way want to speak of this-wavering, irregular, and inexact graph; I evoke by it the ideal form of the diagonal and of the square. I eliminate the empirical, I dematerialize reasoning. By doing this, I make a science possible, both for rigor and for truth, but also for the universal, for the Universal in itself. By doing this I eliminate that which hides form-cacography, inter­ference, and noise-and I create the possibility of a science in the Uni­ versalfor us. Mathematical form is both a Universal in itself and a Uni­ versal for us: and therefore thefirst effort to make communication in a dialogue successful is isomorphic to the efort to render aform independent ofits empirical realizations. These realizations are the third man of the form, its interference and its noise, and it is precisely because they intervene ceaselessly that the first dialogues are aporetic. The dialectical method of the dialogue has its origins in the same regions as mathematical method, which, moreover, is also said to be dialectical.

To exclude the empirical is to exclude differentiation, the plurality of others that mask the same. It is the first movement ofmathematization, of formalization. there would have to be a different word for every circle, for every symbol, for every tree, an d for every pigeon; and a different word for yesterday, today, and to­ morrow; and a different word according to whether he who perceives it is you or I, according to whether one of the two of us is angry, is jaundiced, and so on ad infinitum. At the extreme limits of empIrICISm, meaning is totally plunged into noise, the space of communication is granular,ll dialogue is condemned to cacophony: the transmission of communication is chronic transformation. Thus, the empirical is strictly essential and accidental noise. The first "third man" to exclude is the em­ piricist, along with his empirical domain. And this demon is the strongest demon, since one has only to open one's eyes and ears to see that he is master of the world.l2 Consequently, in order for dialogue to be possible, one must close one's eyes and cover one's ears to the song and the beauty of the sirens. In a single blow, we eliminate hearing and noise, yision and failed drawing; in a single blow, we conceive the form and we under­ stand each other. And therefore, once again, the Greek miracle, that of mathematics, must be born at the same time-historical time, logical time, and reflexive time - as a philosophy of dialogue and by dialogue.

In Platonism, the link between a dialectical method-in the sense of communication - and a progressive working diagram of abstract idealities in the manner of geometry is not an accident in the history of ideas, nor just an episode in the willful decisions of the philosopher: it is inscribed in the nature of things. To isolate an ideal form is to render it independent of the empirical domain and of noise. Noise is the empirical portion of the message just as the empirical domain is the noise of form. In this sense, the minor Socratic dialogues are pre-mathematical in the same way as is the measurement of a wheat field in the Nile valley.


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When two speakers have a dialogue or a dispute, the channel that connects them must be drawn by a diagram with four poles, a complete square equipped with its two diagonals. However loud or irreconcilable their quarrel, however calm or tranquil their agreement, they are linked, in fact, twice: they need, first of all, a certain intersection of their repertoires, without which they would remain strangers; they then band together against the noise which blocks the communication channel. These, two conditions are necessary to the dia­ logue, though not sufficient. Consequently, the two speakers have a common interest in excluding a third man and including a fourth, both of whom are prosopopoeias of the powers of noise or of the instance of intersection'! Now this schema functions in exactly this manner in Plato's Dialogues, as can easily be shown, through the play of people and their naming, their resemblances and differences, their mimetic preoccupations and the dynamics of their violence. Now then, and above all, the mathe­ matical sites, from the Meno through the Timaeus, by way of the Statesman and others, are all reducible geometrically to this diagram. Whence the origin appears, we pass from one language to another, the language said to be natural presupposes a dialectical schema, and this schema, drawn or written in the sand, as such, is the first of the geometric idealities. Mathematics presents itself as a successful dialogue or a communication which rigorously dominates its repertoire and is maximally purged of noise. Of course, it is not that simple. The irrational and the unspeakable lie in the details; listening always requires collating; there is always a leftover or a residue, indefinitely. But then, the schema remains open, and history possible. The philosophy of Plato, in its presentation and its models, is therefore inaugural, or better yet, it seizes the inaugural moment.

To be retained from this first attempt at an explanation are the expul­ sions and the purge. Why the parricide of old father Parmenides, who had to formulate, for the first time, the principle of contradiction? To be noted here again is how two speakers, irreconcilable adversaries, find themselves forced to turn together against the same third man for the dialogue to remain possible, for the elementary link of human relationships to be possible, for geometry to become possible. Be quiet, don't make any noise, put your head back in the sand, go away or die. Strange diagonal which was thought to be so pure, and which is agonal and which remains an agony.

The second attempt contemplates Thales at the foot of the Pyramids, in the light of the sun. It involves several geneses, one of which is ritual.2 But I had not taken into account the fact that the Pyramids are also tombs, that beneath the theorem of Thales, a corpse was buried, hidden. The space in which the geometer intervenes is the space of similarities: he is there, evident, next to three tombs of the same form and of another dimension - the tombs are imitating one another. And it is the pure space of geometry, that of the group of similarities which appeared with Thales. The result is that the theorem and its immersion in Egyptian legend says, without saying it, that there lies beneath the mimetic operator, constructed concretely and represented theoretically, a hidden royal corpse.

The third attempt consists in noting the double writing of geometry.Using figures, schemas, and diagrams. Using letters, words, and sentences of the system, organized by their own semantics and syntax. Leibniz had already observed this double system of writing, consecrated by Descartes and by the Pythagoreans, a double system which represents itself and expresses itself one by the other. He sometimes liked, as did many others, to privilege the intuition, clairvoyant or blind, required by the first [diagrams] over the deductions produced by the second [words].

There were even, as usual, two schools at odds over the question. One held the Greeks to be the teachers of geometry ; the other, the Egyptian priests. This dispute caused them to lose sight of the es­ sential: that the Egyptians wrote in ideograms and the Greeks used an alphabet. Communication between the two cultures can be thought of in terms of the relation between these two scriptive systems (signa/Cliques). Now, this relation is precisely the same as the one in geometry which separates and unites figures and diagrams on the one hand, algebraic writing on the other. Are the square, the triangle, the circle, and the other figures all that remains of hieroglyphics in Greece? As far as I know, they are ideograms. Whence the solution: the historical relation of Greece to Egypt is thinkable in terms of the relation of an alphabet to a set of ideograms, and since geometry could not exist without writing, mathematics being written rather than spoken, this relation is brought back into geometry as an operation using a double system of writing. There we have an easy passage between the natural language and the new language, a passage which can be carried out on the multiple con­ dition that we take into consideration two different languages, two dif­ ferent writing systems and their common ties. And this resolves in turn the historical question: the brutal stoppage of geometry in Egypt, its freezing, its crystallization jnto fixed ideograms, and the irrepressible development, in Greece as well as in our culture, of the new language, that inexhaustible discourse of mathematics and rigor which is the very history of that culture . The inaugural relation of the geometric ideogram to the alphabet, words, and sentences opens onto a limitless path.

This third solution blots out a portion of the texts. The old Egyptian priest, in the Timaeus, compares the knowledge of the Greeks when they were children to the time-worn science of his own culture. He evokes, in order to compare them, floods, fires, celestial fire, catastrophes. Absent from the solution are the priest, history, either mythical or real, in space and time, the violence of the elements which hides the origin and which, as the Timaeus clearly says, always hides that origin. Except, precisely, from the priest, who knows the secret of this violence. The sun of Ra is replaced by Phaethon, and mystical contemplation by the catastrophe of deviation.

We must start over - go back to those parallel lines that never meet. On the one hand, histories, legends, and doxographies, composed in natural language. On the other, a whole corpus, written in mathematical signs and symbols by geometers, by arithmeticians. We are therefore not concerned with merely linking two sets of texts; we must try to glut: two languages back together again. The question always arose in the space of the relation between experience and the abstract, the senses and purity.

Can you imagine (that there exists) a Rosetta Stone with some legends written on one side, with a theorem written on the other side? Here no language is unknown or undecipherable, no side of the stone causes problems; what is in question is the edge common to the two sides, their common border; what is in question is the stone itself.

What separates the Greeks from their possible predecessors, Egyptians or Babylonians, is the establishment of a proof. Now, the first proof we know of is the apagogic proof on the irrationality of V2 [square root of 2].

And so, legends, once again. Euclid's Elements, Book X, first scholium. It was a Pythagorean who proved, for the first time, the so-called irra­tionality [of numbers]. Perhaps his name was Hippasus of Metapontum. Perhaps the sect had sworn an oath to divulge nothing. Well, Hippasus of Metapontum spoke. Perhaps he was expelled. In any case, it seems certain that he died in a shipwreck. The anonymous scholiast continues: "The authors of this legend wanted to speak through allegory. Everything that is irrational and deprived of form must remain hidden, that is what They were trying to say. That if any soul wishes to penetrate this secret region and leave it open, then it will be engulfed in the sea of becoming, it will drown in its restless currents."

Legends and allegories and, now, history. For we read a significant event on three levels. We read it in the scholia, commentaries, narratives. We read it in philosophical texts. We read it in the theorems of geometry. The event is the crisis, the famous crisis of irrational numbers. Owing to this crisis, mathematics, at a point exceedingly close to its origin, came very close to dying. In the aftermath of this crisis, Platonism had to be recast. The crisis touched the logos. If logos means proportion, measured relation, the irrational or alogon is the impossibility of measuring. If logos means discourse, the alogon prohibits speaking. Thus exactitude crumbles, reason is mute.

Hippasus of Metapontum, or another, dies of this crisis, that is the legend and its allegorical cover in the scholium of the Elements. Parmenides, the father, dies of this crisis-this is the philosophical sacrifice perpetrated by Plato. But, once again, history : Plato portrays Theaetetus dying upon returning from the the battle of Corinth (369), Theaetetus, the founder, precisely, of the theory of irrational numbers as it is re­ capitulated in Book X of Euclid. The crisis read three times renders the reading of a triple death: the legendary death of Hippasus, the philo­ sophical parricide of Parmenides, the historical death of Theaetetus. One crisis, three texts, one victim, three narratives. Now, on the other side of the stone, on the other face and in another language, we have the crisis and the possible death of mathematics in itself.

Given then a proof to explicate as one would a text. And, first of all, the proof, doubtless the oldest in history, the one which Aristotle will call reduction to the absurd. Given a square whose side AB = b, whose diagonal AC = a.


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We wish to measure AC in terms of AB. If this is possible, it is because the two lengths are mutually commensurable. We can then write A C/AB = a/b. It is assumed that a/b is reduced to its simplest form, so that the integers a and b are mutually prime. Now, by the Pythagorean theorem: a2 = 2b2• Therefore a2 is even, therefore a is even. And if a and b are mutually prime, b is an odd number.

If a is even, we may posit: a = 2c. Consequently, a2 = 4c2. Consequently 2b2 = 4c2, that is, b2 = 2c2• Thus, b is an even number.

The situation is intolerable, the number b is at the same time even and odd, which, of course, is impossible. Therefore it is impossible to measure the diagonal in terms of the side. They are mutually incommensurable. I repeat, if logos is the proportional, here a/b or 1/V2, the alogon is the incommensurable. If logos is discourse or speech, you can no longer say anything about the diagonal and V2 is irrational. It is impossible to decide whether b is even or odd.

1) What does it mean for two lengths to be mutually commensurable? It means that they have common aliquot parts. There exists, or one could make, a ruler, divided into units, in relation to which these two lengths may, in turn, be divided into parts. In other words, they are other when they are alone together, face to face, but they are same, or just about, in relation to a third term, the unit of measurement taken as reference. The situation is interesting, and it is well known : two irreducibly diferent entities are reduced to similarity through an exteriorpoint ofview. It is fortunate (or necessary) here that the term measure has, traditionally, at least two meanings, the geometric or metrological one and the meaning of non-disproportion, of serenity, of nonviolence, of peace. These two meanings derive from a similar situa­ tion, an identical operation. Socrates objects to the violent crisis of Callicles with the famous remark: you are ignorant of geometry. The Royal Weaver of the Statesman is the bearer of a supreme science : superior metrology.

2) What does it mean for two numbers to be mutually prime? It means that they are radically different, that they have no common factor besides one. We thereby ascertain the first situation, their total otherness, unless we take the unit of measurement into account.

3) What is the Pythagorean theorem? It is the fundamental theorem of measurement in the space of similarities. For it is invariant by variation of the coefficients of the squares, by variation of the forms constructed on the hypotenuse and the two sides of the triangle. And the space of similarities is that space where things can be of the same form and of another size. It is the space of models and of imitations. The theorem of Pythagoras founds measurement on the representative space of imitation.  

4) What, now, is evenness? And what is oddness? The English terms reduce to a word the long Greek discourses: even means equal, united, flat, same; odd means bizarre, un­ matched, extra, left over, unequal, in short, other. To characterize a number by the absurdity that it is at the same time even and odd is to say that it is at the same time same and other.

Conceptually, the apagogic theorem or proof does nothing but play variations on the notion of same and other, using measurement and com­ mensurability, using the fact of two numbers being· mutually prime, using the Pythagorean theorem, using evenness and oddness.

It is a rigorous proof, and the first in history, based on mimesis. It says something very simple : supposing mimesis, it is reducible to the absurd. Thus the crisis of irrational numbers overturns Pythagorean arithmetic and early Platonism.

Hippasus revealed this, he dies of it-end of the first act.

It must be said today that this was said more than two millennia ago. Why go on playing a game that has been decided? For it is as plain as a thousand suns that if the diagonal or v'2 are incommensurable or ir­ rational, they can still be constructed on the square, that the mode of their geometric existence is not different from that of the side. Even the young slave of the Meno, who is ignorant, will know how, will be able, to construct it. In the same way, children know how to spin tops which the Republic analyzes as being stable and mobile at the same time. How is it then that reason can take facts that the most ignorant children know how to establish and construct, and can demonstate them to be irrational? There must be a reason for this irrationality itself.

In other words, we are demonstrating the absurdity of the irrational. We reduce it to the contradictory or to the undecidable. Yet, it exists; we cannot do anything about it. The top spins, even if we demonstrate that, for impregnable reasons, it is, undecidably, both mobile and fixed. That's the way it is. Therefore, all of the theory which precedes and founds the proof must be reviewed, transformed. It is not reason that governs, it is the obstacle. What becomes absurd is not what we have proven to be absurd, it is the theory on which the proof depends. ["An apagogic proof is one that proceeds by disproving the proposition which contradicts the one to be established, in other words, that proceeds by reductio ad absurdum. -Ed.] Here we have the very ordinary movement of science: once it reaches a dead-end of this kind, it immediately transforms its presuppositions.

Translation : mimesis is reducible to contradiction or to the undecidable. Yet it exists; we cannot do anything about it. It spins. It works, as they say. That's the way it is. It can always be shown that we can neither speak nor walk, or that Achilles will never catch up with the tortoise. Yet, we do speak, we do walk, the fleet-footed Achilles does pass the tortoise. That's the way it is. Therefore, all of the theory which precedes must be trans­ formed. What becomes absurd is not what we have proven to be absurd, it is the theory as a whole on which the proof depends.

Whence the (hi)story which follows. Theodorus continues along the legendary path of Hippasus. He multiplies the proofs of irrationality. He goes up to 07. There are a lot of these absurdities, there are as many of them as you want. We even know that there are many more of them than there are of rational relations. Whereupon Theaetetus takes up the archaic Pythagoreanism again and gives a general theory which grounds, in a new reason, the facts of irrationality. Book X of the Elements can now be written. The crisis ends, mathematics recovers an order, Theaetetus dies, here ends this story, a technical one in the language of the system, a historical one in the everyday language that relates the battle of Corinth. Plato recasts his philosophy, father Parmenides is sacrificed during the parricide on the altar of the principle of contradiction; for surely the Same must be Other, after a fashion. Thus, Royalty is founded. The Royal Weaver combines in an ordered web rational proportions and the irra­ tiona:Is; gone is the crisis of the reversal, gone is the technology of the dichotomy, founded on the square, on the iteration of the diagonal. Society, finally, is in order. This dialogue is fatally entitled, not Geometry, but the Statesman.

The Rosetta Stone is constructed. Suppose it is to be read on all of its sides. In the language of legend, in that of history, that of mathematics,that of philosophy. The message that it delivers passes from language to language. The crisis is at stake. This crisis is sacrificial. A series of deaths accompanies its translations into the languages considered. Following these sacrifices, order reappears: in mathematics, in philosophy, in history, in political society. The schema of Rene Girard allows us not only to show the isomorphism of these languages, but also, and especially, their link, how they fit together.6 For it is not enough to narrate, the operators of this movement must be made to appear. Now these operators, all constructed on the pair Same-Other, are seen, deployed in their rigor, throughout the very first geometric proof. Just as the square equipped with its diagonal appeared as the thematized object of the complete intersubjective relation, formation of the ideality as such, so the rigorous proof appears as such, manipulating all the operators of mimesis, namely, the internal dynamics of the schema proposed by Girard. The origin of geometry is immersed in sacrifical history and the two parallel lines are henceforth in connection. Legend, myth, history, philosophy, and pure science have common borders over which a unitary schema builds bridges.

Metapontum and geometer, he was the Pontifex, the Royal Weaver. His violent death in the storm, the death of Theaetetus in the violence of combat, the death of father Parmenides, all these deaths are murders. The irrational is mimetic. The stone which we have read was the stone of the altar at Delos. And geometry begins in violence and in the sacred." [Hermes]

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PostSubject: Re: Michel Serres Thu Jul 14, 2016 12:31 pm

The Birth of Geometry.

Serres wrote:
"What Thales Saw. . .

Hieronymus informs us that he [ThalesJ measured the height ofthe pyramIds by the shadow they cast, taking the observation at the hour when the length of our shadow equals our height.

- Diogenes Laertius

The height of a pyramid is related to the length of its shadow just as the height ofany vertica4 measurable object is related to the length ofits shadow at the same time of day.

- Plutarch


As Auguste Comte says: "In light of previous experience we must acknowledge the impossibility of determining, by direct measurement, most of the heights and distances we should like to know. It is this general fact which makes the science of mathematics necessary. For in renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics. Geometry is a ruse; it takes a detour, an indirect route, to reach that which lies outside immediate experience. In this case the ruse is the model : the construction of the summary, the skeleton of a pyramid in reduced form but of equivalent proportions. In fact, Thales has dis­ covered nothing but the possibility of reduction, the idea of a module, the notion of model. The pyramid itself is inaccessible; he invents a scale, a type of ladder.  

To measure the inaccessible consists in mimicking it within the realm of the accessible.

The point is to transpose some unreachable figure into a more immediate realm in the form of a miniaturized schema.

Accessible, inaccessible, what does this mean? Near, distant; tangible, untouchable; possible or impossible transporting. 'Measurement, sur­veying, direct or immediate, are operations of application, in the sense that a metrics can be used in an applied science; in the sense that, most

often, measurement is the essential element of application ; but primarily in the sense of touch. Such and such a unit or such and such a ruler is applied to the object to be measured; it is placed OIl. top of the object, it touches it. And this is done as often as is necessary. Immediate or direct measurement is possible or impossible as long as this placing is possible or is not. Hence, the inaccessible is that which I cannot touch, that toward which I cannot carry the ruler, that of which the unit cannot be applied. Som.e say that one must use a ruse of reason to go from practice to theory, to imagine a substitute for those lengths my body cannot reach: the pyramids, the sun, the ship on the horizon, the far side of the river. In this sense, mathematics would be the path these ruses take.

This amounts to underestimating the importance of practical activities. For in the final analysis the path in question consists in forsaking the sense of touch for that of sight, measurement by "placing" for measure­ ment by sighting. Here, to theorize is to see, a fact which the Greek language makes clear. Vision is tactile without contact. Descartes knew this, just as he understood better than most what measurement is. The inaccessible is at times accessible' to vision. Can one measure visually the distance to the sun, to the moon, to a ship, to the apex of a pyramid? This is the whole story of Thales-, who discovered nothing but the precise virtues of the human gaze, just as, somewhat later, Berkeley organized in

an erudite manner a spectacle of light beneath his microscope, a rigorous organon of optical representation. Since he cannot use his ruler, he sets up lines of sight or, rather, he lets light project them for him. As far as I know, even for accessible objects, vision alone is my guarantee that the ruler has been placed accurately on the thing. To measure is to align ; the eye is the best witness of an accurate covering-over. Thales invents the notion of model, of module, but he also brings the visible to the tangible. To measure is, supposedly, to relate. True, but the relation implies a transporting: of the ruler, of the point of view, of the things lined up, and so on. In the realm of the accessible, the transporting is always possible: i n the realm o f the inaccessible, vision must take care o f displacements : hence the angle of sight, hence the cast shadow. Measurement, the problem of relation ; sight, the cast shadow ; in any case, the essential element is the transporting.

Let us return to the schema, that of Diogenes or Plutarch. It deals with things in motion or at rest. The constant factor is the pyramid, immobile for ten centuries beneath the Egyptian sky. The apparent movement of the sun, the length and the position of the shadow are all variable. Everyday experience tells us that the latter depend on the former. Hence the initial idea : the clock. The pyramid is a gnomon and the line of its shadow tells time. Measurement and the gauging of the shadow's varia­ tions mark the rhythm of the sun's course. The gnomon, stable and arbitrary, is only an intermediary object; the variances echo one another.

The goal is either civil or astronomical. With a sundial, the measurement of space only measures time. The sundial, whose origin is lost somewhere at the dawn of time,S will disappear during the quarrel of the Ancients and the Moderns, a quarrel particularly acerbic in regard to clocks. Hence, in Diogenes and Plutarch, the remains of what was once the problem of time: to wait for the moment of equality between an object's shadow and its height, or to observe the two shadows at the same time of day; to keep the sun and its daily course in mind,  

Thales's idea (for we must give it a name) consists simply in turning the process around, that is, in considering and then resolving the reverse problem of the gnomon. Instead of letting the pyramid speak of the sun, or the constant determine the scale of the variable, he asks the sun to speak of the pyramid; that is, he asks the object in motion to provide a constant flow of information about the object at rest. This ruse is much more clever than the one we described earlier: the constant is no longer what gauges the regular intervals of the variable ; on the contrary, Thales gauges, within the variable realm, the stable unknown of the constant. Or rather, with the gnomon, whoever measured space also measured time. By inverting the terms, Thales stops time in order to measure space. He stops the course of the sun at the precise instant of isoceles triangles; he homogenizes the day to obtain the general case. And so do Joshua and Copernicus. Hence it hecomes necessary to freeze time in order to conceive ofgeometry Once the gnomon has disappeared, Thales enters into the eternity of the mathematical figure. Plato will follow him. This is the old Bergsonian conclusion.

How did geometry come to the Greeks? A practical genesis: build a reduced model, have a notion of the module, bring the distant to the immediate. A sensorial genesis: organize a visual representation of that which defies physical contact. A civil or epistemological genesis: take astronomy as a starting point, reverse the question of the gnomon. A genesis that is either conceptual or esthetic: erase time in order to measure and master space. Exchange the functions of the variable and the invariable. The origin of geometry is a confluence of geneses. We must follow the other affluents.

We have said that the essential element is the transporting. For even if measurement can be exact or precise, only the relation is rigorous: the reference of a giant schema to a reduced model. These initial geneses are acts of transporting: reduction, the transition from touching to seeing, and vice versa, the reversal of the gnomonic function, the exchange of the stable and the variant, the substitution of space for time.

in case I wish to recall Thales's theorem, the story of the pyramid can serve as a mnemonic device. In a culture with an oral tradition, story takes the place of schema, and theater equals intuition. The diagram of the theorem can only be transmitted in written form, but, in an oral culture, drama is the vehicular form of knowledge. Myth then, the mythical tale, is less a legend of origin than the very form of transmission; it does not bear witness to the emergence of science so much as it communicates an element of science. Here mathematics is the key to history, not the contrary. The schema is the invariant of the tale instead of the tale being the origin of the schema. To know, then, and, in this case, to know Thales's theorem is to remember the Egyptian tale. To teach the theorem is to tell the pseudo-myth of origin. We know that all mythical tales are merely the dramatization of a given content. Only the mathematical decipherment of the text can demonstrate the relationship of the implied schema to the mobilization which turns it into a trans­missible tale.

What Thales or anybody else perceives cannot be anything other than objects of the same form but of different dimensions. The perception of three pyramids is developed within the space of similarities, and this space is constituted by choice in this place: each pyramid is different and yet the same, like the triangles in Thales's theorem. Hence the story is perfectly faithful to the concept, and similar to the idea of similarity. It is more a question here of tech­ nology than of perception: similarity is the secret to the triple edifice, the secret to its construction. The pure knowledge implicit within the design of the pyramids is certainly homothetic.

Henceforth the entire question of the relationship between the schema and history, of the relationship between implicit knowledge and the artisans' practice, will be posed in terms of shadow and sun, a dramatization in the Platonic mode, in terms of implicit and explicit, of knowledge and practical operations : on the one hand, the sun of knowl­ edge and of sameness; on the other, the shadow of opinion, of empiricism, of objects.

These first two readings reveal convincingly the implicit knowledge that a fabricated object hides within itself.

The thing which is dif­ ficult and ultimately inextricable, which we attempt indefinitely to render explicit without being able to explain it completely, and which is thus forever clouded over, is the modality, the "how" of this implication, which, in an actual application, is clearer. The articulatory mode of luminous knowledge and blind practice is blinder in implication, more luminous in application. The origin of knowledge acquired through everyday practice is on the side of shadow; the origin of a practice ac­quired through knowledge is on the side of light.

In the initial technical activity, knowledge is in shadow, and we are also in the dark as acting beings, trying to situate theory in light.

The technique of measurement which is still a ruse of application, or, as Auguste Comte says, an indirect path, repeats the implication but does not explain it. Thales extracts a technique from a technique, and from a practice he gets another practice. The homology of repetition eventually designates the homothesis, but in each instance within the gangue of the applied. The theory expressed by shadows remains in shadow.

There is no longer any original miracle: different techniques give rise to other ones and perpetuate themselves in repetition; measurement and architecture see the theorem differently, that is all. And we remain in the immense shadow of the secret. For, again, one cannot conceive of the origin of technique except as the origin of man himself, faber as soon as he emerges, or rather, emerging because he is faber. Technique is the origin of man, his perpetuation and his repetition. Hence Thales repeats his very origin, and our own : his mathematics, his metrics of geometry, repeats in another way (and as simply as possible) and designates in another way the modality of our technical relationship to objects, the homology of the fabricator to the fabricated. His mathematics takes its place in the open chain of those utterances and designations, but it does not provide the key to the cipher; it does not excavate the secret articula­ tion of knowledge and practice in which the essential element of a possible origin is located. His mathematics is the relation between two shadows, two secrets, two forms and two ciphers, relation or logos, relationship and utterance to be transmitted, utterance which transmits a relationship. As is commonly said, it measures the problem, takes its dimensions, poses it, weighs it, demonstrates it, relates it, but never resolves it. The logos of shadows is still the shadow of the logos.

Still, what Thales's mathematics recounts, at its very inception, is the de-centering of the subject of clear thought with regard to the body that casts its shadow: the subject is the sun, placed beyond the object, on the other side of the shadow. This was also Copernicus' lesson. What this mathematics articulates is the Platonic decision that a geometry of mea­ surement is but a propaedeutic. What it announces, for the first time, is a philosophy of representation, dominating both the pure diagram and its dramatization beneath the torches of the solstice.

Now how is one to study and learn about a volume if not by means of a planar projection? And how can one lay hold of it if not by attacking its surfaces? Thales's geometry says this, and so do architectural technique and the mason's daily practice. In each of the three cases it is a matter of studying a solid in terms of all the bits of information that have been gathered at the relevant levels: the secrets of an object's shaded surfaces and its cast shadow. I know nothing about a volume except what its planar projections tell me. But a projection assumes a point of view and a drawing on a smooth surface, a surface without any shaded area and without any hidden fold.

The cast shadows vary, the secrets are transformed, but they share among them a secret which remains constant and which is the unknown, the pyramid's secret: its inaccessible height. As variable as representation may be, it still designates, suddenly, a portion of the real, a stability proper to the object, its measurement. Which is why, from this position, I can only know about the volume that which is said, written, or described by cast shadows - the bits of informa­ tion transported onto the sand by a ray of sunlight after its interception by the angles and summit of an opaque prism. This geometry is a per­ spective (an architecture), it is a physics, an optics: the shadow is a black specter.

The theater of measurement demonstrates the decoding of a secret, the decipherment of a writing, the reading of a drawing. The sand on which the sun leaves its trace is the screen, the wall at the back of the cave. Here is the scene of representation established for Western thought for the next millennium, the historically stable form of contemplation from the summit of the pyramids. Thales's story is perhaps the instauration of the moment of representation, taken up ad infinitum by philosophers, but also and above all by geometers, from Descartes and his representational plane to Desargues and his point of view, from Monge and his descriptive diagram to Gergonne and his legislative transfers:8 the first word of a perspectival geometry, of an architectural optics of volumes, of an in­ tuitive mathematics immersed in a global organon of representation, the first instance of the Ptolemaic model of knowledge. But from Thales's time to the present day we have forgotten that the shadow was cast, transported by some supporting device, that it itself transported certain information.

The most important question - which messenger transports (and how?) which message?-was covered over for centuries by the f)linding scenography of the shadow-light opposition.

Thales's story is not unanalogous to that of Desargues: the size of the stones, a perspectival geometry, the theory of shadows.9 Nor, after all, to Plato's stories: the sun of the same, the other and empirical object, the cast shadow of the shaded surface , similarity, the cave of representation . Is it a tale of origin ? Yes, and in several ways : the origin of a technology, of an optics, of a philosophy of representation. Of a geometry? Per­ haps-if geo-metry is that triangulation which Plato scorned for being pre-mathematical . It is a mnemonic recipe, friend of the cultural memory because of its forceful dramatization and mythification under the sun of Ra, easy to transmit within a homogeneous cultural setting,IO the ruse of applied mathematics, of an architect and of an expert builder. Even Descartes, followed by Desargues and Monge, remains in the domain of applied geometry as well as that of representation; they perpetuate an engineering geometry that is metric and descriptive. They exhibit the archaic forms of pre-mathematics that run through history.

Like Thales, they impede the formation of pure mathematics. And the latter will emerge as soon as this geometry dies-very recently. And HusserI will write The Origin of Geometry as the bell tolls its disappearance, as if an immense historical cycle had finally come to an end. Thales's story tells something like the story of the birth of a geometry, the measured division of the earth and the differences in shadow and light written on the earth by solid figures and the sun; it does not tell of the birth of mathematics. As proof, let us cite Plato, who, in order to bring about this miracle, requires something else : the essential reality of idealities. Question : how can the pyramid be born as an ideal form?

Every­ thing transpires as if Plato had relegated Thales's story to the depths of his cave. The flat, even wall is always bright: on it the volume casts a shadow; light creates a shaded area. My knowledge is limited to these two shadows; it is only a shadow of knowledge. But there is a third shadow of which the two others provide only an image, or a projection,

and which is the secret buried deep within the volume. Now it is probable that true knowledge of the things of this world lies in the solid's essential shadow, in its opaque and black density, locked forever behind the multiple doors of its edges, besieged only by practice and theory. A wedge can sunder the stones, geometry can divide or duplicate cubes, and the story, indefinitely, will begin again; the solid, whose surfaces cannot be exhausted by analysis, always conserves a kernel of shadow hidden in the shade of its edges. Thales, while readi!1g and noting the volume's traces, deciphers no secret except that of the impossibility of penetrating the volume's arcana, in which knowledge has been entombed forever, and from which the infinite history of analytical progress bursts forth as if from a spring. In this case, his history tells the conclusion of a story, that of the confrontation with solid objects, that of the attack on compact volumes, comprehended like theoretical, objective, unconscious elements, like theoretical, objective, indefinite unknowns. In this case the thing exists qua thing, like an unknown and a correlate, like a secret involuted into thousands and thousands of replicas. Two decisions: either I recognize the object by its shadow, which gives rise to geometry or, better yet, to the idealism of representation, or I allow for a kernel of shadow within the object. In the latter case, theory and practice develop this secret infinitely in a perpetually open history, the history of science, which admits that the solid always envelops something that can be rendered explicit. In Plato, the idealism of representation appears re­ pressed in the depths of his cave, and realism is assumed. However, the story begun in the Nile delta will soon be completed by a sudden and incredibly audacious coup d etat: the radical negation of interior shadows. The Sun of Thales and Ra, the sun whose rays are shut out for an im­ peccable definition,ll is reduced to the meager fire of the prisoners of representation. Thales's theorem, schema of this story, is in the cave's shadow. Outside, the new sun gives off a transcendent light which pierces things and transmits an all-seeing vision. This is how the marvelous miracle is accomplished : the transparency of volumes, the metaphorical naming of the realism of idealities. From the cave to the world outside, the scenography turns into an ichnography : the shadow of solids played on the plane of representation and defined them by boundaries and partitions; now light goes through them and banishes the interior shadow. In place of a planar triangulation of geo-metry there is now a stereometry of empty forms in the epiphany of diaphanousness. The archaic Thales of mensuration gives way to pure geometry, pure because it is cut through by the intuition of transparency and emptiness. Then and only then can the pyramid be born, the pure tetrahedron, first of the five Platonic bodies. By this miracle the sun is in the pyramid: the site, the source of light, the object, all in the same place.

Beneath this new sun, solids no longer have a shadow or a secret; light passes through them without being interrupted, just as it glides along a straight line or a plane; the world they constitute is thoroughly knowable. One can understand the importance that Plato and his school constantly attribute to the stereometry of volumes. The open history of infinite explicitations is closed by this power move, by this stroke of lightning that rips away the veils of shadow; this history is reoriented toward the transcendency of forms. There is no more specter, or analysis; the three shadows (the one on the shaded area of the surface, the one cast, and the one buried within) are snatched away by the sun of the Good. And, as if to close the circle in all rigor and for the coherence of global history, the Timaeus will constitute the world by means of these five bodies : the first, the simplest, the tetrahedron in fact, will be fire. Plato has the pure pyramid come into existence beneath the fires of the sun, and from this tetrahedron he has fire born again: a double miracle that fulfills the scriptures, the Egyptian legend, and the initiation of intuition by posi­ tioning the source of light within the polyhedron. When the pyramid is itself fire (did its name influence its legend?), the sun passes through it.

The entire myth of origin, even that of The Republic, is thus immersed in a vision of fire and dramatizes a solar rite. The new Thales can no longer see any shadow beneath the furnace that pure form and the solar hearth constitute: original conjunction of mathematical stereometry and the mythical element, blinding atmosphere of the first philosophies of intui­ tion. The kernel of knowledge is continually enveloped by myth, and the myth is ceaselessly generated within the theater of representation. The­ory, vision; light, fire. We have here a new genesis with four branches where two tributaries are mixed: science and the history of religions. From astronomy to solar mythology.

Nevertheless, this power move is not exactly a revolution. Plato kills the hen that laid the golden eggs: by cutting through the solids he nullifies history ; the eternity of transcendency freezes the diachrony and the genealogy of forms.

The realism of transparent idealities is still immersed in a philosophy of representa­ tion. Of course, ichnography is substituted for scenography, but the former is a trans-representation from a divine point of view. To go beyond Thales's scene, the shadowless theater is still a theater. The inevi­ table realism is still an idealism: the geometric form clearly expresses this difficulty. This form is pre-judged to be without shadow or secret, it exists itself and in itself, but it never hides anything that could exceed the definition one has fixed for it. It exists as an ideality, transparent to vision, transparent to noesis. It is a theoretical element known thoroughly, something seen and known without residue. Intuition is blinded by its existence, but intuition passes through it. Its identity guarantees that it is ubiquitously identical, and hence its perception is not interrupted. Vision and knowledge are white specters. Now, precisely when this pure ge­ ometry, inherited from Plato, dies, when it is no longer possible to assume intuitive principles, when the theater of representation is closed, the secret, the shadow, and the implication will explode again among these abstract forms before the eyes of dumbfounded mathematician s - ex­ plosions that had been announced before all these deaths throughout history. The right angle, the plane, the volume, their intervals and their areas, will be recognized as chaotic, dense, compact - again teeming with folds and dark hiding places. Pure and simple forms are neither that simple nor that pure; they are no longer complete, theoretical knowns, things seen and known without residue, but rather theoretical, objective unknowns infinitely folded into one another, enormous virtualities of noemes, like the stones and the objects of the world, like our stone con­ structions and our wrought objects. Form hides beneath its form trans­-infinite kernels of knowledge which, one might fear, history will never exhaust; these highly inaccessible instances become our new tasks. Mathematical realism is weighed down and takes on the old density that Plato's sun had dissolved. Present-day mathematics, although maximally abstract and pure, is de­veloping in a lexicon that derives in part from technology. It is a new way of listening once more to Thales's old Egyptian legend.

What Thales saw at the base of the pyramids (the sun, the homo­ thetic edifice, the shaded surface and the cast shadow), what Thales did alongside the pyramids (the partitioning off and the measurement of similar triangles in the parallelism of two gnomons, one of which is our body), are the thousands and thousands of implications that the history of science is slowly developing and that the eternal geometers will see, without always seeing them, and will create, without always knowing it. These implications express nothing less than the obscure articulations of rigorous knowledge and the totality of other human activities, indefinitely abandoned to their obscure fate." [Hermes]

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"All that exists is just and unjust and equally justified in both." [Aeschylus, Prometheus]

"The history of everyday is constituted by our habits. ... How have you lived today?" [N.]

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PostSubject: Re: Michel Serres Thu Jul 14, 2016 12:37 pm

Dynamics from Leibniz to Lucretius: The Abstraction of Physics

lIya Prigogine and Isabelle Stengers wrote:
"The Systeme de Leibniz begins by positing the major problematic of Serres's work: what is at stake in the hypothesis of the great classical rationalist who supposes that the passage from local to global is always possible? As we shall see, the question of the integrable world authorizes dreams of determinism. What we shall call here the classical style in physics is expressed in Laplace's dream of a world made of determinist and calculable trajectories. Laplace's demon observes the instantaneous state of the world and integrates its trajectory. He thus has access to both the past and the future in the minutest detail. This dream of omniscience translates Leibniz's baroque monadology using the unidimensional plati­ tude characteristic of the nineteenth century.

Nine years later, in La Naissance de fa physique dans fe texte de Lucrece, Serres takes up, fulfills, and modifies the project sketched out in Le Systeme de Leibniz: the confrontation of the rationalism of differential and integral calculus (Leibniz) with the rationalism of ancient atomism.

We observe that, as a good Epicurean, Lucretius answers "no" to the following question: "Is the passage from local to global always possible?"

This distinction between the pure and the impure crippled Leibniz's reputation as a physicist. Though his role in mathematics is recognized, in physics he figures more often than not as the inopportune and obstinate adversary of Newton, the person whose ambition to create communicative paths between physics and metaphysics led down the road to perdition.

Thus Leibniz's rigor was judged severely. It is commonly agreed that, though he was the creator of the term "dynamics," he nevertheless "missed" the mathematical physics created by Newton at that very mo­ ment. This is explained by the fact that for him philosophical rigor came before the needs of an inductive and necessarily approximate science. He refused to give up the idea of the rational nature of the real, measured not by the yardstick of man, who observes and generalizes, but by that of God, who, calculating, created the world. Thus Leibniz was unequivocally a "pre-Newtonian." This is a condemnation, moreover, that is sufficiently justified by his rejection of the principles of inertia and of interaction at a distance-in short, of Newtonian physics. In the face of this condemna­ tion, we can make three remarks.

In the first place, one might well ask, solely based on the facts, whether it is not the history of physics that has "missed" Leibniz. The discovery of his role and influence will undoubtedly offer a few surprises when the history of Continental physics is better known: Bernoulli, Euler, and D'Alembert were neither Newtonians nor pre-Newtonians. Second, the role played by God in Leibnizian physics does not allow the opposition of this physics to Newton's in the same way that metaphysical speculation might be opposed to positive scientific praxis. Think of the controversies between Leibniz and Clarke, Newton's proxy, or of Newton's own con­ siderations on the production of forces of attraction by the active prin­ ciples that show the action of God on the world :  these will suffice here as examples. Certainly Laplace (the "second Newton") was not wrong to say to Napoleon that his system had no need for the hypothesis of God: this remark merely expressed the fact that God, in the role assigned to him by Newton, did not resist the progress of dynamics. Last, it cannot be denied that, if not since Newton, then at least since Laplace, we have accepted the systems of interactions at a distance as part of our physical world.s But one must distinguish between such a conception and the de­ velopment of "mathematical physics," that is to say, the creation of the formalism that is today called dynamics. We have said, and we wish to show, that the language of dynamics has, in a certain sense, changed from a Newtonian to a Leibnizian one. The world of trajectories determined by forces can henceforth be thought of as being identical to the Leibnizian system of the world in which every point locally expresses the global law.

To introduce this thesis we propose some little-known satiric verses of James Clerk Maxwell, which are doubly interesting because they celebrate both what we call the "Leibnizian" transformation of dynamics and some­ one who was among the first to explore the possibilities and powers of a role that was then new within the scientific community, that of "textbook writer " :

But see! Tait writes in lucid symbols clear One small equation;

And Force becomes of Energy a mere Space-variation.

Force, then, is Force, but Mark you ! not a thing, Only a Vector;

Thy barbed arrows now have lost their sting, Impotent spectre !

Thy reign, O Force! is over. Now no more Heed we thine action ;

Repulsion leaves us where we were before, So does attraction.

Both Action and Reaction now are gone. Just ere they vanished,

Stress j oined their hands in peace , and made them one ; Then they were banished.

The universe is free from pole to pole, Free from all forces.

Rejoice! ye stars-like blessed gods ye roll On in your courses.

The stars and, following their example, all physical bodies travel through the universe like free and self-determined gods, each following its own law. Newtonian physics posits a body assumed to be isolated, endowed with a rectilinear and uniform inertial movement, and calculates the modifications of this movement as determined by the action of forces. For Leibniz, the forces are not "given" and are in no way the real causes of the modification of a movement but rather are local properties within a dynamic system: at every point, they characterize a momentary state belonging to a series regulated by a law.!

In the same way, since Lagrangell and especially since Hamilton, mathematical physics has abandoned Newtonian representation. Instead of calculating the action of each force on each point, it first of all proposes the system in its canonic form, and constructs a function (the Hamiltonian in particular, a representation of energy - see Maxwell's versesfthat de­ fines the global state of the system. From this function, the set of "forces" acting on each point at every moment can be derived. Forces are no longer responsible for accelerations; rather, they are deducible from the structure of the dynamic system defined by the Hamiltonian; they are the effects of the global law of evolution which the Hamiltonian ex­presses.

What we would discover is deducible from the fact that the dynamic problem is henceforth posited in the same way that Leibniz posited it: movement is produced within a full world, an interdependent world in which nothing can happen that has not been made possible by the state of the set of bodies according to a harmony that determines and checks at every moment the unfolding of the different movements. What Leibniz thought of as a preestablished harmony translated at every instant by the conservation of energy the physics of Lagrange put to work through the study of movement as the succession of states of equilibrium, disrupted and reestablished at every instant, and Hamiltonian formalism transformed into an a priori syntax of the formal language in which every dynamic problem can be posited. During the eighteenth century, in fact, physicists succeeded in inscribing Newtonian physics as a special a. posteriori case within the a priori conceptual framework of Leibnizian physics.

We know that for Leibniz the physics of aggregates of bodies affecting each other had only an imaginary character. It is a dream (but a coherent one) to attribute the variations of force at each point to external factors. That is to say, in reality no longer from a physical point of view but from a metaphysical one, the world is composed of unextended substantial elements-monads-each of which displays a predetermined internal law:

"The monad automatically deciphers, both in itself and for itself, a universe that is at once its closed interior, its own account of it, and the extensive entirety of its exteriority. . . . The monad is full to an inaccessible extent of attachments that are sufficient for representing  a full and compact nature like itself: the inherence of all the numbers belonging to deciphering. Impressed and expressive but never im­ pressionable."

It is God alone in the Leibnizian system who can know monads as such; it is he who, at the origin, calculated each one's individual law so that the monads express each other, so that each one, through its internal law, translates every change that has occurred in another-that is to say, so that the universe described by physics is imaginary but not illusory.

But the jump from the imaginary point of view of the monad dreaming itself and dreaming of things affected from the outside to the point of view of God-from the physics of aggregates to monadology-has now received a purely physical translation: every integrable system, every system whose equations of movement can be integrated, allows for a monadic repre­ sentation.

Let us first of all define what we mean by an integrable system. A problem put in the canonic language of dynamics is presented in the form of a set of differential equations that describes the following situation for every point: at every instant, a set of forces derived from a function of the global state (such as the Hamiltonian, the sum of kinetic and potential energies) modifies the state of the system. The.refore this function as well is modified : from it, a momentlater, a new set of forces will be derived . To resolve a dynamic problem is, ideally, to integrate these differential equations and to obtain the set of trajectories taken by the points of the system.

It is evident that the complexity of the equations to be integrated varies according to the more or less judicious choice of the canonic variables that describe the system. That is why dynamic physics as it was formulated in the nineteenth century is a theory of transformations among canonic languages, among points of view on a system, each de­ scribing this system in terms of a different set of canonic variables and thereby placing it within a different space defined each time by these variables. More precisely, it is a theory that allows for the choice of the best point of view so that the system can be integrated and the trajectories calculated.

But what integration could be easier than that of the movement of an isolated body, with no interaction with the rest of the world? No ex­ ternal perturbation determines a change in velocity, which thus remains constant, while the position is a linear function of time. All the energy is kinetic; the value of "potential energy" is zero.

The optimal point of view on a system, the best choice of variables, is therefore the one that cancels out the potential energy redefined in terms of these variables. And dynamic theory tells us that every integrable system can be represented in this way-can be redefined as a set of "units" evolving in a pseudo-inertial movement, without any interaction among the "units." Each "monadic unit" is no longer determined in each of its movements by interactions with the aggregate ; each deploys its own law for itself, alone in a system which it reflects intrinsically, because its very definition supposes and translates this system in every detail. There is full passage between the local and the global.

Michel Serres has shown that Leibniz did not "speak of" science, did not "speak about" it from an external position; he "spoke" science, and did so even when he "spoke" metaphysics. Thus, speaking the language of dynamics in a philosophical manner-moreover, a language which was based on his work in physics-Leibniz arrived at a conclusion which was only to be found by dynamics in its most abstract state. This is not an anachronism; Leibniz is not even a "precursor"; he introduces no new fact or concept. He simply undertakes-with the rigor for which he is criticized - an exploration of that internal coherence of the physical and mathematical language of his age which he contributed to creating. The point of view he attributes to God is a privileged point of view whose existence is affirmed by physics as soon as the system can be exactly integrated.

But-and this is how the difference between classical dynamics and Leibnizian metaphysics is now defined-we know that the class of in­ tegrable systems is extremely restricted (the theorem of Liouville). More­ over, our formalized science (we shall soon speak of this) is no longer limited to classical dynamics. The world described by the science of irreversible processes, the world described by quantum mechanics, is a world in which interactions play an ever more important role. The Leibnizian exploration of classical style therefore allows us to identify precisely what is now at stake in science: the description of a world of processes, the definition of entities that participate in the becoming of the world.

We have not shown that the Leibnizian system could be entirely reduced to a theorem oj dynamics, nor that the plurality of isomorphic languages could be reduced to one: the language of dynamics is only one model among others in the Leibnizian system. We have simply shown the pertinence of this model. In fact, for the past two centuries, there have been physicists who affirmed that the whole world could be described as if it were an integrable dynamic system : this is what we have called the Laplacean dream.

Let us examine the properties of the dynamic world based on.the model of integrable systems, which is also' the world whose legality Lucretius's clinamen will undermine. It is a world of determinist and re­ versible trajectories whose definition presupposes two disparate kinds of information: knowledge of the law of evolution which syntax allows one to formulate a priori from the definition of the forces of interaction and the binds inside the system, and knowledge of the description of any state of the system. From this point on, "everything is given." The law will lay out the trajectory taken both toward the past and toward the future. The law is generality itself: it defines the limits of all the possible evolutions of the system and defines them as equivalent to each other, each reflecting the arbitrary particularity of an initial condition.

The property of reversibility is given in a very simple manner: the law of dynamics is such that the operation of instantaneous inversion v -> -v of velocities at each point of the system is equivalent to the operation of inversion of the direction of the flow of time t -> -t. For any dynamic evolution it is thus possible to define an initial state (in fact, the one prepared by this operation of inversion of velocities) such that the system undergoes inverse evolution, "moving in reverse."

This property is an excellent illustration of the arbitrary and de­ terminist character of dynamic evolution. Generally, for any given state, the law permits the calculation of the initial condition needed for the system so that it ends up "spontaneously" at a specified moment at that state. In this world of automata, both arbitrary and inflexible, to know is in fact to dominate; the most extravagant evolutions are deployed in­ differently, translating the extravagance of an initial state. Among these extravagant evolutions, "moving in reverse" has the force of a symbol. Everyone knows the absurd impression provoked by movies shown in reverse: burning matches which become reconstituted; flowers which become buds; a wave of water in a swimming pool which projects a diver up onto the springboard. Dynamics describes and postulates this absurd world.

The notion of reversible and determinist trajectories does not belong exclusively to classical dynamics. It is found in relativity and in quantum mechanics; the evolution of the wave-function as defined by Schrodinger's equation also echoes the syntax of dynamics.

Rather unexpectedly, it is in quantum mechanics that the monadic character of every integrable system has been most evident. In Bohr's model of the atom each orbit is characterized by a well-determined energy level in which electrons are in steady, eternal, and invariable movement. The steady state of orbital electrons is the typical example of the monadic state. The orbits are defined as being without interaction with each other or with the world ; it is as though they were isolated, alone in the world. This monadic description was absorbed in the modern formulation of quantum mechanics by means of Schrodinger's equation : this description becomes a privileged representation such that quantum evolution is reduced to the evolution of a set of isolated steady states without inter­ actions which remain identical to themselves for an indefinite time.

One might object that we have not spoken of the second half of Bohr's model: electrons can jump from one orbit to another, emitting or ab­ sorbing a photon of energy corresponding to the difference of energy between initial and final levels. For that reason we can know that these levels exist; the electron without interaction is unknowable.

It is here that quantum mechanics decisively parts company from dynamics; quantum formalism does not define the determinist and re­ versible description as being complete. It associates a second type of evolution with it, one that is irreversible and discontinuous, the reduction of the wave-function that corresponds, for example, to the jump from one orbit to another with the recorded emission or absorption of a photon, or corresponds to any other interaction with an instrument of measure­ ment, after which one can deterministically attribute a numerical value to one of the quantum parameters. Thus irreversibility of the measure­ ment is necessary to the definition of the quantum phenomenon. Starting from quantum mechanics, the physicist knows that there are interactiom in the physical world which cannot be eliminated by a dynamic transfor­ mation. The process that ends with the amplification and recording of a quantum phenomenon at the macroscopic level is one such interaction, as is the world of unstable sub-quantum particles.

Quantum mechanics thus presents a reversal of perspective relative to classical style. It is no longer a question of looking for simplicity at the level of elementary behavior. Dynamic simplicity, as reflected by the possibility of a completely monadic representation, belongs, in fact, to the macroscopic world, to the world on our scale. Our physics is a science created by macroscopic beings, created with conceptual tools and instru­ ments that belong to the macroscopic world. It is from that position, when we question the world of quanta, that we must choose what will allow us to express matters in terms of measurable, reproducible, and communicable properties. We can no longer allow ourselves, as far as the physical world is concerned, the privileged point of view which, when pushed to its limit, we once could have identified as that of God.

Wte can se how unfortunate was the widespread assumption that quantum mechanics "discovered" that the process of measurement disturbs the system nwasured. uncontrollably modifying; the values of certain parameters in order to ascertain the value of others. Such an assumption in fact implies that only an arbitrary positivistic prohibition prevents us from speaking; of "hidden variables," that is to say, prevents us from affirming; that the system in question is, at every moment, defined by the set of physical parameters, even if all of them cannot be known simultaneously.

The actual situation is entirely different. The real discovery of quantum mechanics, as it is expressed by the inseparable character of reversible evolution and irreversible reduction, is not that tht, process of measurement disturbs, but rather that it participates in the defini­tion of, the measured parameter, so that this parameter cannot be attributed to the quantum system "in itself" and one cannot speak of "hidden variables." As Niels Bohr repeatedly said, quantum mechanics discovered the necessity of choice, choosing; what question to ask, in other words, choosing; both the instrumental fram('work of the qu('stion and on(' of the complementary descriptions articulated among; themselves by formalism but irreducible to a single description.

If dynamics is above all a science of the macroscopic world, the fol­ lowing question immediately comes to mind : in the natural world, where irreversibility seems to be the rule, what is the status of the reversible

descriptions of dynamics? Two solutions have often been proposed, and the ground common to both is denial of the existence of the problem. According to the first, reversible description is only an idealized and partial model that must be abandoned or "completed" adequately as soon as it is no longer valid. According to the other, irreversibility is only an illusion ; we are dream-like automata "swimming" in one direction in the sea of an eternal and legal world.

Neither of these solutions, too simple and especially too sterile, has ever really taken hold. The first amounts to the acceptance of a strict separation of disciplines and to the idea that a style is never more than an inflated paradigm. The second, in contrast, brings us back to a pseudo­ Leibnizian style, telling us that the irreversible world is only a well­ fabricated illusion determined by our subjectivity, and that objective reality is reversible, legal, and determinist.

But a style is not abandoned in the name of prudence and plausibility. Like the Laplacean dream, repeatedly declared defunct, it regroups and deploys, changes arenas, rises again under different theoretical guises. On the other hand, our style is no longer the classical style : from Laplace to Du Bois-Reymond, the nineteenth century, which made dynamics the basis of a conception of the world, was also the period in which a new history and culture arose, and thus a new style of science.

"What is the Industrial Revolution? A revolution operating on matter. It takes place at the very sources of dynamics, at the origins of force. One takes force as it is or one produces it. Descartes and Newton, crowned by Lagrange, chose the first alternative: force is there, given by the biotope, the wind, the sea, and gravity. It is beyond our control except insofar as men and horses are subject to it, but it is not under our dominion when it is a question of heavy bodies, of air, and of water. With it one produces motion, work, by using tools. . . . Then a sudden change is imposed on the raw elements: fire replaces air and water in order to transform the earth . . . . Fire finishes off the horses, strikes them down. The source, the origin of force is in this flash of lightning, this ignition. Its energy exceeds form; it trans­ forms. Geometry disintegrates, lines are erased; matter, ablaze, ex­ plodes; the former color-soft, light, golden-is now dashed with bright hues. The horses, now dead, pass over the ship's bridge in a cloud of horsepower."

From mechanics to thermodynamics, changes of style and society have occurred. And, in parallel fashion, dynamics developed and reached a point of formal perfection. For the last century we have been faced with an original scientific problem, that of the articulation of styles and of reversible and irreversible time. We no longer live in Leibniz's time: to speak the language of dynamics, and conclude from it that we are dreamt monads, is for us not to speak "science" but to speak "a science," not only against the style of the age and against eventual personal beliefs, but also against another science.

Michel Serres has often cited Leon Brillouin's response to the claims of dynamics: a dynamic description is only determinist if the description of its initial state is completely accurate - and accuracy is expensive. All the energy in the world could not pay the energetic debt of a completely determinist description on a global scale. In this form, the argument is perfectly correct: no description actually produced will be perfectly de­ terminist. However, determinism stands out as a limit, perhaps an inac­ cessible one in practice but one which nevertheless defines the series of increasing precision: style resists the argument of plausibility.

However, dynamics has discovered today that as soon as the dynamic system to be described is no longer completely simple, the determinist description cannot be realized, even if one dismisses questions of cost or of plausibility. In other words, we now know that there are dynamic systems of different sorts. There are the rare ones in which determinism exists as a limit-state, costly but conceivable, in which extrapolation is possible between the approximate description of any observer and the infinitely precise one of which Leibniz's God is capable. And there are systems in which the idea of determinist prediction conflicts with the laws of dynamics and in which the idea of determining the initial con­ ditions becomes unthinkable. In certain cases, the passage between local, dynamic descriptions and global vision is impossible.

Since the end of the nineteenth century, dynamics has had a new history, born of two necessities: one, coming from astronomy, the need to define exactly the trajectories that determine the interactions among more than two bodies, and the other, the need to derive from dynamics the description of irreversible evolution, typically defining the increase

in entropy tending toward a maximum. The first led to non-integrable systems, the second, to research on complicated dynamic systems which would not a priori exclude complicated evolutions (ergodic systems, mixing systems, etc.). The history of dynamics has therefore been marked by the coexistence of the two styles; the fruit of this coexistence, the formulation of a broadened dynamic theory in which an "operator" of entropy can be defined, does not belong to history but rather to current research in physics.

In fact, dynamics here encounters the mixture : the systems which, since Poincare, have been known not to be exactly integrable and the systems which are studied by statistical mechanics cloud the view of the observer, even the demon. The science of analysis and of separation must hence­ forth, as Serres says, becalm itself, feminize itself, erase itself, with observation disappearing in favor of relation : "The world as it is is not the product of my representation; my knowledge, on the contrary, is a product of the world in the process of becoming. Things themselves choose, exclude, meet, and give rise to one another."

The systems which aim for the theorem of impossibility of which we are speaking are called unstable systems. In order to understand what an unstable dynamic system is, let us describe a stable system. A rigid pendulum can have two kinds of movements that are qualitatively dis­ tinct : oscillation and rotation. The behavior of a pendulum is predictable and depends on the initial conditions. There is only one case of uncer­ tainty; it occurs when the initial acceleration is such that the pendulum attains the vertical with zero velocity. In that case, a perturbation "as small as one wishes" will be sufficient to determine which side it will fall to and thus what kind of movement it will adopt. The pendulum is thus the type of system for which, with the exception of these individual rare cases of uncertainty, an approximate description is sufficient to avoid any unexpected evolution and for which a determinist description is the limit. An unstable system, on the other hand, is a system in which the initial conditions determining various qualitatively distinct behaviors are not clearly separated but are, on the contrary, as close as one might wish. We are all familiar with this sort of intimate mixture - it is described by number theory: every rational number is surrounded by irrationals, and every irrational by rationals. Similarly, whatever the neighborhood defined for an initial state, one always finds at least one other state giving rise to a qualitatively different behavior, just as oscillation and rotation are qualitatively different.

Under these conditions, in order to predict deterministically the type of behavior the system will adopt, one would need infinite precision. It is of no use to increase the level of precision or even to make it tend toward infinity; uncertainty always remains complete-it does not diminish as precision increases. That means that divine knowledge is no longer implied in human knowledge as its limit, as that toward which one might tend with increasing precision ; it is something other, separated by a gap.

For the third time in the twentieth century, physics finds itself defined by the fact that we cannot observe and measure with positively infinite precision, no more than we can communicate faster than the speed of light or measure with instruments that are not macroscopic. Just as the demonstrations of impossibility in relativity and in quantum mechanics are tightly linked to the opening of a new conceptual field, the impos­ sibility of conquering the indeterminacy essential to unstable dynamic systems is not an epistemological discovery which only concerns the re­ lation of our knowledge to the world; rather, it offers a new method of positing problems of physics: the possibility of positing the problem of irreversibility within dynamics. It is not a question of recognizing that we are incapable of calculating such trajectories; rather, it is a question of realizing that the trajectory is not an adequate physical concept for these systems. Henceforth the field of dynamics will appear larger: systems described in terms of trajectories with their determinist and reversible properties are only a particular class within that field.

Where monadic physics ends and trajectories become unstable, the world of the irreversible begins, the open world in which, through fluctu­ ations and bifurcations, things are born, grow, and die. The instability of trajectories, their irreducibie and essential indeterminacy, have as a global result the heavy, macroscopic irreversibility of the self-organizing processes that make up nature.

To speak of unstable trajectories is to use the language of classical dynamics to introduce non-classical physics; it is to use the determinist and reversible model to construct the description of the irreversible; it is to invoke monads to describe interactions that cannot be reabsorbed in the monadic interior. In short, it is to repeat Lucretius's procedure, "starting" with the inflexible and legal order and then introducing dis­ turbance and indeterminacy. Things are born where the law is not suf­ ficient to exclude disturbance or to prevent the dynamic monads from interacting.

"Without the declination, there are only the laws of fate, that is to say, the chains of order. The new is born of the old; the new is only the repetition of the old. But the angle interrupts the stoic chain, breaks thefoederafati, the endless series of causes and reasons. It disturbs, in fact, the laws of nature. And from it, the arrival or life, of everything that breathes ; and the leaping of horses."

The model of falling atoms, parallel flows in an infinite void, eternally identical to itself, constitutes an exact image of the monadic evolutions of classical dynamics, parallel as well, without interactions, in a reversible, that is to say, in an indifferent, world. The fall is nothing but the uni­ versal without a memory whose every instant is the integral repetition of the preceding instant. Classical dynamics was the mathematico-physical effectuation of this ordered world, directed by a law.

But the parallel flow is only one of the models of the primitive base from which things are born. The second is that of the cloud, of stochastic chaos, closer to reality, Serres says. Here, atoms go in all directions and collide randomly; it is an immense, tumultuous population, a "disor­ ganized, fluctuating, brownian mass composed of dissimilarities and op­ positions." Here as well one must note the exactness. This is a descrip­ tion of another physical situation, a purely macroscopic one but one which also is integrally subject to a law: statistical equilibrium within a population in which all the processes and their opposites are produced simultaneously and compensate for each other. At this macroscopic level, just as at the microscopic level of the description of atomic trajectories, the apparent absurdity of the clinamen is repeated. Statistical disorder should not produce a difference, any more than a disturbance should occur in the established trajectory, for disorder is the state in which all differences are abolished in the indifferent, senseless tumult in which they all coexist.

Yet non-classical science has taught us that trajectories can become unstable and that stochastic chaos can become creative. In certain cir­ cumstances, evolution bifurcates, the homogeneous disorder is no longer stable, and a new order of organized functioning is established, with amplified fluctuation. For example, laminar flow in parallel sheets "spontaneously" becomes turbulent; that is to say, we now can calculate it. In this realm of the bifurcation, in which turba becomes turbo, rather a strange tumult reigns, the complete opposite of indifferent disorder. Creative chaos is illegality itself, for its description dissolves the distinc­ tion between the macroscopic state and the microscopic fluctuation; correlations can appear among distant events; local deviations echo through­ out the system - the matrix-state in which fluctuations are amplified and from which things are born.

And thus Serres is correct: the question is reversed. It is no longer necessary to ask where the clinamen comes from or how one might justify the disturbing of laws. All laminar flows can become unstable past a certain threshold of velocity, and that was known just as the productive nature of organized forms, of bifurcating evolution, of what we call dissipative structures, was known. One must ask how an abstraction of this knowledge could have been made to describe the world in order, subject to a universal law. We already know one answer given by Serres. Classical science is a science of engineers who knew, of course, that their flows were never perfectly laminar, but who made the theory of laminar flow perfectly controllable and directable, the only flow for which know­ ing is controlling.

"The technological model is in place. It is a physics of water mains. Our physics was first of all a physics of fountain-builders, of well­ diggers, or of builders of aqueducts. . . . Hence this physical world where the drainpipe is essential and where the clinamen seems to be freedom because it is precisely the turbulence that refuses enforced flow. Incomprehensible by scientific theory, incomprehensible to the hydraulic engineer."

The history of science particularly repeats itself when it is a question of mastery or control. We could rewrite the same text with irreversible replacing clinamen and the builder of thermic machines replacing the well-digger, with matter transformed by fire and heat replacing flowing water. This text would tell of the birth of the thermodynamics of equi­librium, the classical science from which will later be born the study of irreversible processes." [Excerpted in Serres, Hermes]

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PostSubject: Re: Michel Serres Fri Jul 15, 2016 9:32 pm

Serres is a Girardian.

His book 'Rome' is an attack on the blond beast, and the barbarous founding at the heart of the Empire.

He like Girard sees the scapegoat victim, as the centre around which a mob organizes itself into a Form.
Violence as the foundation.
Romus kills Remus.
Patrocles dies in stead of Achilles.
The King is annually felled like the sun-king, and so is twin to the scapegoat, the pharmakos.
Violence, as Girard noted, grows acute with the growing erasure of difference - twins, twin figures, and eventually the mob where one is just as well as any other.

The scapegoat is a ritual substitute, and therefore a proxy.
He likens this to the Joker card in a deck that by itself having no value, is capable of taking any value, capable of being anything and everything - from king to clown to victim…
It dons many masks.
The place-holder 'Joker' changes the series of relation.
The law of circulation makes possible a body - the king's body, a sovereign body.

From him, we may appreciate the word 'Vicar', which helps differentiate I.E. notions of the Druid or the Priest.

Vicar:

From Old French vicaire "deputy, second in command," also in the ecclesiastical sense (12c.), from Latin vicarius "a substitute, deputy, proxy," noun use of adjective vicarius "substituted, delegated," from vicis "change, interchange, succession; a place, position". The original notion is of "earthly representative of God or Christ;" but also used in sense of "person acting as parish priest in place of a real parson" (early 14c.)."

Vicarious:

"1630s, "taking the place of another," from Latin vicarius "that supplies a place; substituted, delegated," from vicis "a change, exchange, interchange; succession, alternation, substitution," from PIE root *weik- (4) "to bend, wind" (source also of Sanskrit visti "changing, changeable;" Old English wician "to give way, yield," wice "wych elm;" Old Norse vikja "to bend, turn;" Swedish viker "willow twig, wand;" German wechsel "change")."


The neutral symbol that can take on any signifier is a proxy, a ritual substitute, the king-And-the-scapegoat…
It circulates like a phantom taking on forms.
Substituting for the absolute absence, the Vicar who himself acts as a "proxy of God", Constructs and Founds the sacred around the ritual substitute, the scapegoat, the pharmakos.

In the sociological sense, this means, against Hobbes, the social is a formation not between man and man, but between the collective and the coveted/dispelled object.

Serres' view is as Xt. as Girard's, where everyone is constituted and mutually implicated through every other.
It de-centers identity - which is what postmodernism is.

Still, there is much to appreciate in all of Serres' books.

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PostSubject: Re: Michel Serres Fri Jul 15, 2016 9:32 pm

Serres wrote:
"Livy 1.7
Romulus and Remus, abandoned Alban twins, suck at the dry breast of the she-wolf [louve]; I say dry because the Latin for she-wolf, lupa, indicates a whore, a brothel whore [une putain de lupanar].
False sons of a whore, true sons of a vestal and Mars, legendary, sons of violence and rape, sons of the god of war and a chaste and savage priestess, Romulus and Remus are also grandchildren of enemy brothers.
Murder among brothers didn’t begin today.
So Romulus kills Remus, and he founds Rome.

Before this history and at about the same spot, a legend is recounted. One twin, Romulus, has just executed his twin, right here, and he is sacrificing. To Hercules in particular.
Hercules had passed through there. Heracles, the twin of Iphicles, had killed Cacus there. As though a murder always preceded a murder. As though one foundation weren’t sufficient to truly begin. As though an origin required its origin.
No doubt we will never know if there’s a single word of truth in all this, truth in the sense of naive history; no doubt we will never know whether myth, softly, came down to earth, the earth of phenomena. Did Livy know? Livy is hesitant; he recounts cautiously; he reports traditions.

But suddenly everything changes. From the beginning, from Aeneas’ departure after the capture of Troy, it has been a question of Rome, of its founders and their long genealogy, and now here abruptly is a legend, here is a narrative within a narrative, well defined, like a cartouche in a painting, like a legend at the bottom of a map. It has been a question of men, Aeneas, Latinus, Romulus, and now here is a god, Hercules. It was an urban place; and now here is a pasture, meadow grass for cattle. Latin was being spoken, and now Greek is being spoken: Geryon, Cacus, Evander. Suddenly the proper names are Hellenic, and the atmosphere is divine.

Gods come before kings. A hero becomes a god in the spot where the twin becomes a king.
Hercules mounts an altar, Romulus a throne. Romulus killed Remus, Hercules killed Cacus. Romulus risked his life in the battle to the death in the midst of the mob. Hercules risked his life in the crowd of shepherds from the vicinity, all come to help Cacus. Hercules was recognized to be a god, a son of a god by Evander; Romulus seeks recognition, he seeks legitimacy. The legend intervenes in the middle of the legendary narrative with a change of language, a change of register, tone, with a change of scale, state, space and time; you might say a metalanguage.

Romulus, remaining alone, is sacrificing. But he is sacrificing to Hercules, among other Alban rites. Hercules comes into the Latin narrative as foreign. Foreign due to his Greekness, due to his warrior function during the reign of the first king. He has just dispatched Geryon during a triple combat and stolen his cattle, wonderfully beautiful. Tired, he rests in the thick grass. But while he is sleeping, Cacus, a shepherd from thereabouts, wants to steal his cattle and hide them in his cave. Livy seems unaware of Cacus’ three heads, but that’s of no importance. The latter, profiting from the night, leads the most beautiful of the animals into his cave, dragging them by the tail; in this way their tracks are only turned toward the outside. At dawn, Hercules, awake, looking for his cattle, is fooled by the stratagem: confused and uncertain, he prepares to leave the disquieting place. Right then, the remaining cattle of the herd low, and the cattle hidden in the cave low in response. Hercules returns, wields his club and beats Cacus to death. Evander, arriving, pardons this murder; the king recognizes the god and promises him a temple. End of the episode.

Evander, writing, I mean the human, the historical, are never deceived: they set up an altar, a temple, at the murder site, the greatest of altars even, the Altar Maxima. And this was the first sacrifice, where the cattle were killed in Hercules’ stead.
Evander who transforms the murderer into a god, who discredits Cacus.  
Evander, a Greek, writes in a single direction. He changes writing like a good semiconductor; he forgets one of the two directions; he transforms the alternating into direct.

We must then go back along the steps of the cattle, enter into the black cave where Cacus lies. It’s pitch black in there. The victim ox is silenced, a knife at the throat; Cacus is mute, dead and put at fault. Remus is beneath Cacus, Cacus is beneath the ox, lowing since the dawn.
From this black cave, the black box of origins, origins that are present, absent and concealed beneath the writing of this Latin which signifies concealed, from this black box, meaning, writing, tracks, raw lowing, sound, voice, history, origin shoot out, burst forth, in every direction [sens]. The detailed account of time and its conditions.

Cacus is a thief; he is Hercules’ victim; Hercules is a murderer; Cacus is a poor shepherd defended by the shepherds of the vicinity; Hercules was very nearly lynched by them. Everything comes from that black box, the cave with the cattle, the den full of wildlife, the mêlée, the black mob of the three Curiatii around the young Horatius, and whoever is in the box can be black or white. Box: the mêlée of the collective.

Rome is founded on a murder, and this murder refers to another murder. He who digs the foundation finds a head at the Capitol, a body, a skeleton, a mass grave. He finds Cacus beneath Remus. And so on. He who enters into a foundation enters a tomb; Rome is the city of tombs.

To every question concerning each person, the answer is that he has played every role. Each of them is at once murderer, victim, powerless witness to the thing, fair, liar, silent and unaware, good and bad, medium, mixed, grey or mediocre, man, hero, shepherd, king, coward, ox and god, just like me and you.

History opens amid these virtual bouquets.

And what if myth, from the origins of the world to our time, had obscurely played this role?

Let’s begin again: Hercules is a robber, Hercules is robbed; he is a murderer, he is going to be murdered; he is deified. He takes on every value: base man, hero, god. Cacus is a robber, Cacus is robbed; just as much robber and robbed as the hero himself; confident in his strength just as much as him; Cacus is murdered; Cacus was slandered by the base name attached to him; he is perhaps accused in order to justify Hercules and his apotheosis; Cacus is a quasi-Hercules, Hercules is a super-Cacus. Evander the supermale, the just and learned man, cheats and lies. What comes out of the black box is a thousand-voiced language, what comes out of the black box has a thousand substitutive meanings [sens]. The truth is never anything but a stability amid substitutions; it is only an invariant amid their changing.

The one that goes from Mommsen to Corneille, from Shakespeare to Dionysius, and from annalistic to fasti.
The immense software body of the city. Analysis rages; criticism and hypercriticism, dispute occupy the terrain; it’s the new war of all against all. Everyone takes his part of it and considers the part of the others to be unfounded, crude and unformed. History has taken the place of empire. We have changed all that, and we have passed from hardware to software. We no longer take cities; we no longer take anything but names. But the murderous rule has nonetheless been maintained. The goal is to mark one’s name and erase the others. Customs have passed from murder by sword to murder by language. The passage from the thing to the sign doesn’t fundamentally change anything. We are all Coriolanus: taking the city by taking its name. And we don’t hear the lamentations of the women. The historians become agitated, during the storm, around the Roman corpus.

Lamentation. Why must we still tear bodies to pieces? Why must we tear the corpus apart? Why must we always destroy the objects of knowledge?

Answer: in order to found. Silvia must be buried, Remus lynched, Romulus torn to pieces, to found the city. This is how it was founded. This is how. This is how knowledge is founded. This is how the institutions of knowledge are founded. The accord of the scholarly collective is isomorphic to the analysis into fragments of the corpus. Each scholar had a piece of the known in his robe, his discipline. He rules in his empire department. He rules in his reason. This is how Rome was founded. So be it. This is how the institution of knowledge is founded. So be it. This is how the collective is founded, and whichever it may be, urban, academic, etc. This is the true conflict of the faculties. Lamentation: what good is it to seek to know this? Why open the black box? Why unfold the hem of the toga?

Answer: in order to found. We now know how to found. We can found a new city, a new knowledge, a new history. Do we want to? Do we want to cut up, analyse, tear apart the king’s body again?

What do we want to found, what city do we want to found? That’s the only question. What good is all of philosophy and all of history, what good are lives of work at the break of dawn without an answer to this question?

The answer is clear and simple. We no longer tolerate the thanatocracy or its acts, its discourses or the culture it brings about. The hatreds it spreads, the disputes, polemics, divisions, powers, resemble each other. The culture founded on death only returns to death, repetitively. The knowledge founded on murder returns to murder multiplied by knowledge. This multiplication was strong enough recently to absorb all that exists. Today we are at the point of no return. We have returned to the time of foundation, crushed by the thanatocracies of irrationality and rationality. We want to avoid a new repetition of the identical ancestral gesture. A city, science or knowledge that would no longer be founded, like ours, on death and destruction remains to be founded. We no longer have anything but this work, apart from the dismal repetitions of history.

The hatred between brothers goes to point of murder. This hatred between twins grows like ivy in the group; it goes toward extinction, and there is no foundation unless, stopping the reciprocal throat-slitting, the crowd turns against one among them. René Girard’s lesson, the schema he has proposed makes these two contradictory stories compatible; it even makes them complementary. The fight between twins is an operator; the death in the middle of the mob is a point of articulation. The former is the motor force of a time, and the latter is the end of one time and the beginning of another. Hence the reference point for the foundation.

How was Rome founded by the founding family? Romulus disappears in the middle of the Fathers; Remus falls under blows in the middle of the mob. The twins are killed with a twin death. A double tradition is there so as to hide this. The one jumped over the wall, and the other disappeared from the throne during the storm.

How was Rome founded by the founding family? The mother having disappeared, Romulus having been carved into pieces, Remus having fallen beneath the blows of the crowd, the entire family is lynched. The criminal mother, a violated vestal, the brother who is a criminal for having killed his brother, the twin who is a criminal for having overstepped borders.

The victim is the element of substitution. The neutral element. The white element, the one that can bear every value.
The black box is full of white elements.
Thus myth is algebraic.
I’m using ‘white’ in the sense it has in games and in light. White is the sum of colours; it can be broken down into the spectrum of the rainbow. Thus the white token can settle on any value. You are an animal, a shepherd, a king and a god; you are white; you are good and bad, robber and robbed, murderer and murdered; you are white. You are victim, substitute or one substituted for; you are white.

When you aren’t white, a determination appears, a mark or a sign. Determination is negative; if you are king, here and now, you aren’t an ox or a shepherd or a hero. Indetermination is positive. White is the indeterminate, the limit of the underdetermined, the whole of the positive. You are white – yes, yes, yes – you are every possible world. Leibniz had possible worlds be visited in a pyramid; the pyramid is fire, the fire is white light.
In the beginning is the black box, ignorance, our zero of information. In the beginning is the white, every possible world. In the beginning is the victim, this relation of substitution, and this death, between us.
Livy is at the origins; he writes, with his good hand, the book of foundations. Before Rome, there’s dead Alba, a sacrificed white virgin.

Livy 1.16; Plutarch, ‘Romulus’ 27–29
Man is a wolf to man, violence. Man is a god to man, sacred. How can the change between the state posed by Hobbes and the one said by Spinoza be described? The history of Rome and the foundation of Rome answer the question. They anticipate the two principles. Romulus is wolf and god.

I have given the name joker, or white domino, to a kind of neutral or rather multivalent element, undetermined of itself, and which could take on this value, identity or determination or that one, depending on the system of proximities it is inserted into. I can say of the joker that it is a king, that it is a jack, that it is a queen or any other figure. The very first of the Italian algebraists had called this the cosa, the thing; and we have in turn called it the unknown = x. Neither the joker nor the thing are unknown; they are merely undetermined. As is said of the chameleon, they are determined by the environment. Or by a decision external to them. Or by the set in which they are put into play.
This joker can be found everywhere, from mathematics to the social sciences, from the theory of money as a general equivalent to every simple or complicated practice of deciphering.
Thus there are white elements.

Without them, we couldn’t think the multiple, nor could we construct models. That’s because incompatibilities increase very quickly with the terms put into play, their number and their combination. Noise, friction, interceptions, blockages invade the set, whether it be the pure multiple, chaos, or a well-formed system, ready to function. To eliminate the difficulty, Leibniz rigorously eliminated relations as well; the monads, without holes or doors, are blind, mute and deaf, except to God, the supreme exchanger and only centre. Reinject relations into the multiple, and noise, sound irrepressibly return: parasitic interference. Leibniz’s is the only solution to this problem that has been found, except for the lie told everywhere that a new proposed solution would be completely different. It’s always the centre that propagates this lie and that makes enough noise to monopolize noise and impose silence on noise, that behaves enough like a parasite to cause the deaths of the little parasites, that won’t tolerate relations between elements except for the very ones imposed by it. What is quite modestly called the monopoly of legitimate violence. I only know one solution that’s completely different from that one: injecting these jokers, these white elements into the multiple, set or system. Locally they are repressors of incompatibility; they take incompatibility into themselves as their own determination; they therefore bring it to zero. The undetermined play the role of lubricant. The friction rolls over the white elements. As a result, they give the multiple a fuzzy appearance. Fluid.
This solution is new, no lie.

A god is a true joker. The more god he is, the more joker he is. Look at Jupiter: he becomes a swan, he’s a bull, he’s that golden rain that beats on Danaë’s threshold.
The divine is totally substitutable, totally vicarious, totally vicar and victim. Seen from here, Jupiter is a bull, from there, he’s a swan; perceived from the doors of the girl Danaë, he’s ample and abundant golden rain.
Jupiter is the geometral of substitutions; he is the ichnography of the monstrous. The god is not one monster; he’s every possible monster; he sums the scenographies of monsters.
He is, thus, a white element, white like the sum of colours. And this is why a mythical narrative so often has every value; it puts jokers or white elements into play. And this is why it always overhangs the set of explanations, all of them linear and analytical, all of them inclined.

Myth understands history; no history explains myth.
History is an analytical series issuing, like the cattle, from the black box with white elements.
The Iliad is a possible sequence of the infinite games of the Olympic gods; the history of Rome ensues from legend, legend in fact, not myth, since Hercules is the only divine joker in it.
Legend is a myth that’s been slightly inclined. I mean a geometral that’s been slightly scenographied. I mean a compass that’s slightly fixed. It inclines toward the downstream; it descends a little slope.

We are no longer very far from the origins. The black box is Pandora’s box; anything can come out of it.
Myth is rich with everything substitutable; legend includes much of it. Thus Livy, Roman, is closer to history than Homer, Hellenic, but not much closer.

The perfect myth, telling the divine, mixes white elements. It draws a geometral by means of jokers. It obtains then every meaning; it is a sum; it is, if I may use the word, pansemic. So any meaning of history is already understood by it. Theorem: we will always be able to understand history by means of theologies.

Silver and gold, paper money are general equivalents. A sum, a sum of money is a white element.
With it you can get a bull, a lake full of swans, make a rain of gold flow on Danaë’s propylaea, tempt, it seems, the gods themselves. A narrative told by means of such jokers is, once again, a geometral. So any meaning of any history is already understood by it. Theorem: we will always be able to understand history by means of economy.

Conclusion: economy and theology are equivalent explanations of history. Equivalent: I mean omnivalent.
We will sacrifice, in the same temple, to Jupiter and to Quirinus.
But, the geometral is obtained by substitution, by the set of victimary substitutions. Omnivalence is obtained by violence.
We will sacrifice, on the same altar, to Jupiter, to Mars, to Quirinus. Theology, violence and economy are on the same line, or rather, they occupy the same space, I mean: all of space.

All the Fathers, together, carved the king up into pieces.
Each one grabs a member and hides it in his bosom. The one who conceals it is thus marked; he is marked with the king’s seal; he holds the scepter. Look at him, he is discovered now; he was hiding the king’s cut-off hand, an end of a stiff finger, in his hand. So he is recognized. He is recognized as king; he is recognized as guilty. All the Fathers together are marked with the seal. Marked, they are all condemnable; all risk therefore being treated in turn as they treated Romulus. And as they will treat Caesar, the man, precisely, that is torn to pieces, Caesar in diasparagmos.

EMPEROR CAESAR, these two words form a nice black box in the now clear sense of an undivided division, a denied scattering, or the hidden diasparagmos. He who, among the Fathers, is caught carrying that denied part, that empire, he who is convicted of doing so, is in Romulus’ state. A wolf from having killed, a king from carrying the part, a god from being put to death in his turn. They are, together, these hidden parts of Romulus. Each carries the impartitive empire; they are all this empire. Caesar, the son of the Caesar who was torn to pieces, the son of a torn women, of an eviscerated mother. Each for his part is this impartitive Romulus, the murderer of Remus, the wolf twin of the wolf son of a she-wolf, the king of Rome, the god carved up for the apotheosis, the son of Mars, god of violence, and of the violated vestal. Each here is a wolf, the twin of the wolf fathers in the city’s Senate, carved up into some number of elders and conscripts.

Each one, each Caesar, hides his part of the empire; each denies being in possession of it. But everybody suspects everybody and keeps a watch on the fold of the other’s robe. The black of his soul, as they say, or his unconscious. I know what the black fold of your unconscious contains. They hate each other for having torn up the seamless tunic. Each one hates the robe of the other and the fold of cloth, black. The ring formed by the Fathers, the zone around the throne, becomes tighter from this hatred, and loosens as well from this surveillance.

They exchange names; they exchange stones; they all resemble one other without unanimity. The war of all against all is perhaps going to take place, and it assembles them. Nothing makes things stick together better than the glue of resentment. Hatred separates them; it reunites them. War moves them apart, and it brings them back together. All the evil in the world, it is said, is born from suspicion: the war here is born from the suspicion of empire, however this is nothing but an open secret. But what a contract, already, their war, their hatred and their suspicion are; what a pact their squabble is; what a glue their colloquy and their resentment.

Each one knows, by himself, what all the others are hiding. Each one knows, thanks to the others, what he is hiding from himself.
Each one has stolen his part and conceals it; he is thus marked. Not all of them will be discovered at once.
There they are, seeking out like dogs. They sniff like wolves.
Everyone suspects everyone. Who is going to discover who, now? And everything begins again.
Everyone, again, discovers one of them and recognizes him. They recognize him as guilty: Romulus killed Remus, and this new one killed Romulus … They recognize him as carrying the empire. His part hidden in the fold of his robe was impartitive; that which was to be demonstrated.
Power cannot be divided, and, when it is divided up, it is hidden.
The important thing is to open the black box of the toga. What is really there in the fold? That very thing enveloped there and which has no name in any language is repugnant enough that the fold has remained closed. Let’s not be afraid; let’s open the box. What’s in the box? A quasi-object.

I am never alone in relation to an object. My attention, my perception are plunged in a social and cultural set. A theory of knowledge in which the subject, a monad, relates to an object, passive or active, is an empty utopia. The object is constituted in and through the relations of the group. A stake, for the combat of competition, a fetish, for vainglory and prosternation, merchandise, for commercial exchange, it is an object, more rarely. The subject of the object is always multiple.

Conversely, the collective never manages to form without that element I’ve called quasi-object circulating in it, the ball in the team, the peace pipe among the enemies finally reaching an agreement, the common glass at the feast where there is drinking, at the unanimous last supper, the small change at the market. This circulation is necessary for the distributed multiple to become a collective. It’s not constituted by a contract – we don’t know where it’s written. It’s not any will – we never find its subject. It is a token running from body to body, quite simply.
There is no object without a collective; there is no human collective without an object. Rome constructs the object.

The subject, on its own, isn’t at the foundation of knowledge, and the transcendental is not in the subject. Knowledge is nothing without a collective that founds it. The collective is nothing without the circulation of the quasi-object. This circulation is concealed; the quasi-object itself is hidden.
Knowledge is founded in collective practices that the collective is ignorant of. This black and white set of ignorance and knowledge can perhaps be called religion.

No human collective can exist without an object. There is no object without a collective; there is no collective without an object. The political animal is a fiction. The group floats crazily; it only becomes fixed for a time by means of an object. This proposition is no doubt generalizable: the more animals live in collectivity, the more they are forced to construct things outside their bodies, outside their group. These aren’t the goals of their collective; they are its conditions.
The object was truly Rome’s genius. Corneille constructs that object from a decapitated body in The Death of Pompey. Livy and Plutarch construct it slowly, without seeing it, as Rome is being founded.

I have called what circulates in a group and constitutes it through its circulation a quasi-object. It is sometimes hidden; this is the case with the slipper in ‘hunt-the-slipper’. It passes. This is the case with the parts of the royal body, of the imperium, or of the empire. No one knows who is in possession of it, for he hides it; yet everyone is suspected of being in possession of it. It is this passage and this suspicion that constitutes the we, and not the addition of the Fathers, of the players who are present, their set as such.

The quasi-object, this part, is the being of their relation. When it is discovered, its possessor becomes a singularity of the set. He is only one part of it, and he possesses the essence of social relations; he is only one part of it, and he possesses the whole. He holds in hand, in the exact sense, the empire. The quasi-object resolves the problem of the totality without addition: no, it’s not their sum that produces the Senate or the set of the Fathers. It’s the trace of blood on the insides of their togas. The king’s body passes, part after part, passes from black fold to black fold; it traces routes in the black box. It is a tracing element; it reveals the network of unobservable relations in the box. Yet, the one who is discovered sees or can see all the quasi-twins swoop down on him to accuse and lynch him; he will therefore be thrown under them, subject, Caesar, Caesar at the Senate, under the conspirator Fathers. The quasi-object, a totalizing part, or tracing element, is an integrator. It can vary in form, nature or circulation, so the group varies with its variation. But in any case, it is the primary condition that allows an elementary form of society to be born.

This condition is not to be sought in a contract that precedes history. It is not in an abstract concept nor it is in a general will about which no one has ever known whose will it was. It is not in an idea or in a person, in a being or in a subject, in a preliminary writing or state. It is in an object. Neither the cloud nor the thunder has its law in the head or heart of a subject Jupiter, be he transcendent God or metaphysical Nature. Their law is in the relations of a charge. Things are no different here, except that neither the same laws nor the same objects are at issue. This quasi-object is a marker of relations; without it the relations would vanish and disappear. They float in the immediate. The quasiobject stabilizes time. While it is passing, the network is fairly stable. It is the first object of history.

It is the condition, the foundation of the group. As though the depositing of its first time.
It is the condition, the foundation of Rome. As though the impartitive part of the empire. It is a scattered member of Romulus’ body. Analysis and synthesis, dispersion and centralization. The object here crystallizes an energy, sums up a dynamic, the one that disperses the parts and assembles them, the same one that made the people flee and that, through this flight, constructed the encounters, storm and fracas, return to the tranquil and serene light, stones and voices.

Transubstantiation. Romulus, the king of Rome, and Mettius, the dictator of Alba, are torn to pieces, by quartering or another method. Tarpeia and Turnus are visibly stoned. Rhea Silvia no doubt is buried alive.

These murders first of all form a series; the founding family begins; it gives a global image of it: the mother is buried; one twin is lynched; the other is carved up. This series never stops, all the way to Julius Caesar, all the way to the foundation of Christianity and after. It is the current foundation, recurring throughout time; it could be used to delimit eras or periods. It is rarely made visible. The arrogance of our culture suffers from its barbarous foundation.

This series, impressive in its returns whatever the degree of culture might be, is endowed with a simple reason. This reason is a law; this law has meaning. The king, the dictator are divided into parts. They are unities; they are cut up into parts.
The law is quite simple; it follows a star schema: the one becomes multiplicity; the multiple becomes unity. We have to think together lynching and stoning, a mob pouncing together upon a body and execution by quartering. We could even name this law: the law of empire.

The people disperse, and it’s in scattering that people encounter each other. They move away and scatter through thrown rocks; they assemble through voices. What I have called systrophe above is this exchanger, this dynamic converter of the multiple into the one and the one into the multiple, this crucible in which energies spread, expand, that is to say, move and condense anew.

Two things are to be distinguished a bit: the group as such and the quasi-objects that circulate in it.
The group explodes and reforms under the action of these forces, diastole and systole, atomization and moving closer again, and the quasi-objects vary: stones and voices. The stones separate. Stoning: they converge. They are suddenly like the parts of the royal body returned to their place. They will tear another body to pieces as well. The quasi-object, at first an unformed part hidden in the fold of the toga, the quasi-object becomes objectivized,

Blood circulates; they are covered with it; each one of them is or has a member. There is no human collective without an object. Here is the first object. Each of them thus participates in the same body. This object becomes transubstantiated; this means that the relation of the group becomes transformed. This object becomes stone; it becomes bracelet; it becomes shield; it is always related to the object it comes from. It is always fragment; it always comes from unity so as to return to it. It is the tracer, the tracing element, which reveals, in the black box, how the energies that make it a converter move and condense. It is fragment, and it is sign, a hardware fragment and a software sign. It is a fragmented piece: he who is under the pile of fragments has received the suffrages under the fracas of the acclamations. He is stoned in a hardware kind of way; he is elected in a software kind of way. The transubstantiation is there: the stone becomes bread, and the bread becomes body. No. The stone becomes body, and the body becomes sign.

The quasi-object remains quite real; relation is a being among us, and without this being, there, there is no relation, that is to say, no group, but the being of this being evolves and fluctuates. It is flesh, it is stone, it is voice. It goes from real presence to symbolic presence, and that’s another transubstantiation. From flesh to voice or from stone to noise. What must be understood here is a series as well, a series of transubstantiations, of which the one we have canonized is only one stage. Flesh, stone, bread, body, blood, voice, sign, and so on. And changing substance signifies changing being: the being of the relation, the being of a null being, the being of the collective.

A goat in a bush, encountered by chance, always takes Isaac’s place. Taking the place and ceding the place, again, cause time to run. The culprit is endlessly being found; he is endlessly being proclaimed innocent. Substance is another name for substitution. Substance stops; substitution flees.

Kingship, the reign existing from deferring the day of drowning in the middle of the Albula, the day of wrath when the diasparagmos explodes, the day when the crowd pushes the king from the top of the execution rock, the day when divine wrath strikes the king down with lightning; the gods are on the earth that day. The king is nothing but the death of the king; he is the moment of his collective death; he is sacred due to his expulsion; he is sacred due to his lynching; kingship is nothing but the waiting time until this day.

All the power is in the hands of he who can, of he who is going to die at the hands of all. He polarizes the collective violence. He holds the potential monopoly or mono-pole of collective violence. He holds the capacity, the power of the social accord. He doesn’t hold it in the order of the sign, in the order of the contract; he holds it in the real order of energy and force. The energy is directed toward the one, he who is capable of every energy, of the whole of force; he holds this knot, potentially. He waits for this capacity to pass into act. I’m taking the word capable in the geometric sense.

Kingship directly, carnally, savagely, takes on being put to death, which is like paying off the debt for power.

Power is transmitted like a red hot coal, the red flower of the wolves in the depths of the jungle.

Livy measures the distance to the foundation: however far we seem to be from it, history, recounted or real, doesn’t lose sight of it. Distance from the in-stance to the circumstance.

With his title, Livy invites us to measure the distance or interval and to assess a slow moving away. Along this path, then, an abrupt fall stops it and throws everything toward the state of equilibrium; everything returns, one might say naturally, from the environments of said circumstance to the in-stances of the foundation, from the distance taken to that zero state where distance is nil. Rome never ceases being founded; its history or its time is simply what happens between two occurrences of the founding gesture. Here it is freeing itself from it, climbing, rising in relation to its stable seat, sometimes for a long time, then it falls there; it falls back there. Rome begins. Even when it becomes the city of Augustus, it is still a city of auguries; Rome is the city of beginnings. The beginning that we know is simply the time in which many beginnings accumulate.

Endowed with a powerfully attractive black centre in which the beginning is involuted and endlessly restarts, endowed with a deep pole, that of foundation. Rome begins again and is founded; at its beginning it was founded often. Hardly visible little distancings, weak rays, narrow festoons. Its proper time is this false rhythm, in which the brief borders the long, in which the immense is next to the aborted. This time is close enough to that of life, near that of genesis; it’s astonishing that it has long been confused with eternity.

Rome rarely leaves the time of beginning; it endures by returning there. The redundancy sown on the multiple is the return to the foundations.

Rome is mired in the sacred. Its feet are tied up in the snares of the sacred. Blind, its attempts to free itself are visible; it is inexorably brought back. Drugged, intoxicated with violence, it can’t escape the sacred. The eternal return is exactly the return of the sacred, the return of the forgotten, of the buried, the return to the light of the head with the intact face at the top of Capitoline Hill.

Rome always has one foot in this grave, and it falls back in. Return. It leaves it due to the abundant and perennial spring.

The time of invention comes to terms with the time of repetition. The time of newnesses comes to terms with the returns of sacred redundancy. No, the negative doesn’t produce any work, any movement; on the contrary, it annuls all movement by the return to zero, by the eternal return to the act of founding. Killing is nothing but monotonous, and this law produces nothing. The work of the negative is a null work; it nullifies itself at the point of return, at the sacred place of foundation. Real work only comes from diverging from this position, only comes from distance from this return, distance from the fragile disequilibriums around the thesis, only comes from separation in relation to the foundation, in relation to the inexorable invariance of the sacred, with zero information.

Livy holds together, in the palm of his hand and from the origin, the question of the foundation, that of the negative, that of the return, that of the sacred. One little word resolves them: Ab. There is history only in tearing away from there. Rome can do so, Rome can’t do so. Hence the ruleless beating around the thesis. Ab urbe condita. An impaled insect struggles madly around a thorn.

Culture is, quite simply, the distance from this foundation, barbarous." [Rome]

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"ἐδιζησάμην ἐμεωυτόν." [Heraclitus]

"All that exists is just and unjust and equally justified in both." [Aeschylus, Prometheus]

"The history of everyday is constituted by our habits. ... How have you lived today?" [N.]

*Become clean, my friends.*
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